PREFACE 

THE  principal  object  in  writing  this  book  has  been  the 
same  as  that  which  has  governed  the  author  in  writing  other 
mathematical  text-books ;  viz.,  to  bring  out  the  fundamental 
utilities  which  underlie  and  grow  out  of  the  principles  pre- 
sented. Not  only  is  the  fundamental  source  of  new  power 
in  Trigonometry  frequently  emphasized,  but  each  new 
process  is  taken  up,  not  arbitrarily,  but  for  the  sake  of 
the  economy  or  new  power  which  it  gives. 

Among  other  special  features  of  the  book,  the  following 
may  be  mentioned : 

Under  each  case  in  the  solution  of  triangles  two  groups 
of  examples  are  given ;  one  with  the  degree  divided  sexa- 
gesimally,  and  the  other  with  the  degree  divided  decimally. 
The  inclusion  of  the  examples  in  terms  of  the  decimally 
divided  degree  meets  the  new  requirements  of  Harvard, 
Yale,  and  Princeton. 

A  chapter  is  given  on  logarithms  and  their  properties. 
Practical  examples  are  included  in  this  chapter  which  are 
not  only  interesting  in  themselves,  but  which  afford  a  review 
of  and  a  correlation  with  other  branches  of  mathematics. 

When  use  is  made  of  the  line  equivalents  of  the  trigono- 
metric ratios,  it  is  specially  shown  that  such  treatment  is 
merely  a  convenient  substitute  for  the  ratio  treatment,  and 
the  method  of  this  substitution  is  shown  and  its  processes 
carefully  safeguarded. 

A  chapter  is  given  in  which  the  applications  of  trigo- 
nometry are  reduced  to  a  system. 

3 


219350 


4\  i:  ;  f  C- :-  {  : *       :  :\  • . *:    PREFACE 

The  subject-matter  of  the  text-book  is  enlivened  and  made 
more  vital  and  human  by  a  chapter  on  the  history  of  trigo- 
nometry. 

Attention  is  also  called  to  the  method  in  which  logarithmic 
work  is  arranged.  This  form  of  tabulation  is  used,  for 
instance,  in  the  designing  room  in  the  United  States  Navy 
Department  and  by  engineers  in  general.  Among  the  ad- 
vantages of  this  method  of  arranging  logarithmic  work  are 
the  following : 

(1)  It  abbreviates   the  work   by   omitting   the   equality 
marks. 

(2)  It  includes  within  itself  the  actual  numbers  whose 
logarithms  are  being  used. 

(3)  It  facilitates  the  correction  of  mistakes  by  including 
and  presenting  in  order  all  the  steps  of  a  logarithmic  reduc- 
tion. 

(4)  The  arrangement  of  the  work  is  such  that  after  the 
pupil  has  acquired  facility  in  logarithmic  computation,  some 
of  the  steps  in  the  tabulation  may  be  omitted  without  chang- 
ing the  general  form  of  tabulation. 

The  author  wishes  to  express  his  especial  indebtedness 
to  Mr.  Howard  Smith  of  the  Hill  School,  Pottstown,  Pa., 
to  whom  most  of  the  examples  are  due,  and  who  has  made 
important  suggestions  concerning  other  parts  of  the  work. 
The  writer  is  also  under  obligation  to  his  colleague,  Mr.  J.  H. 
Keener,  to  whom  the  examples  in  the  General  Review  Exer- 
cise are  mainly  due.  Professor  William  Betz  of  the  East 
Rochester  High  School,  Rochester,  N.Y.,  Dr.  Henry  A.  Con- 
verse of  the  Polytechnic  Institute,  Baltimore,  Md.,  and 
Professor  William  H.  Metzler  of  Syracuse  University  have  also 
aided  the  writer  by  important  corrections  and  suggestions. 

FLETCHER  DUKELL. 

LAWRENCEVILLE,  N.J.,  January  10,  1910. 


TABLE   OF  CONTENTS 

CHAPTER  I 

PAGE 

LOGARITHMS 7 

CHAPTER  II 
DEFINITIONS.     TRIGONOMETRIC  FUNCTIONS       .        .        .        .        .        .24 

CHAPTER  III 
RIGHT  TRIANGLES 52 

CHAPTER  IV 

GONIOMETRY 73 

CHAPTER  V 
GONIOMETRY  (Continued} 93 

CHAPTER  VI 
OBLIQUE  TRIANGLES 107 

CHAPTER  VII 
PRACTICAL  APPLICATIONS 131 

CHAPTER  VIII 
CIRCULAR  MEASURE.     GRAPHS  OF  TRIGONOMETRIC  FUNCTIONS     .        .     142 

CHAPTER  IX 
INVERSE  TRIGONOMETRIC  FUNCTIONS 152 

CHAPTER  X 
COMPUTATION  OF  TABLES.     TRIGONOMETRIC  SERIES       ....     157 

CHAPTER  XI 
HISTORY  OF  TRIGONOMETRY .     162 


5 


6  TABLE   OF   CONTENTS 

CHAPTER  XII 

PACK 

INTRODUCTION  TO  SPHERICAL  TRIGONOMETRY 185 

CHAPTER   XIII 
THE  RIGHT  SPHERICAL  TRIANGLE 191 

CHAPTER   XIV 
OBLIQUE  SPHERICAL  TRIANGLES 203 

CHAPTER  XV 
SOME  APPLICATIONS  OF  SPHERICAL  TRIGONOMETRY        ....     230 


PLANE  TRIGONOMETRY 


CHAPTER   I 
LOGARITHMS 

1.  The  logarithm  of  a  number  is  the  exponent  of  that 
power  of  another  number,  taken  as  the  base,  which  equals 
the  given  number. 

Thus,  1000  =  103,  hence  log  1000  =  3,  10  being  taken  as  the  base; 
again,  if  8  be  taken  as  the  base,  4  =  8%  hence  Iog4  =  f;  also,  if  5  be 
taken  as  the  base,  log  125  =  3,  log  -^  =  —  2,  etc. 

The  base  used  is  sometimes  stated  in  the  context  as  above ; 
but,  when  desirable,  it  is  indicated  by  writing  it  as  a  small 
subscript  to  the  word  log. 

Thus  the  above  expressions  might  be  written, 

loglo  1000  =  3 ;  Iog8  4  =  i ;  Iog5 125  =  3 ;  Iog5  &  =  -  2 ;  etc. 

In  general,  by  the  definition  of  a  logarithm, 

number  =  (base)logarithm, 
or  N=  Bl ;  hence  logBN=  I 

2.  Uses  or  Utility  of  Logarithms.     One  of  the  principal 
uses    of  .  logarithms    is    to    simplify   numerical   work.      For 
instance,  by  logarithms  the  numerical  work  of  multiplying 
two  numbers  is  converted  into  the  simpler  work  of  adding 
the  logarithms  of  these  numbers.     To  illustrate  this  principle 
we  may  take  the  simple  case  of  multiplying  two  numbers 
which  are  exact  powers  of  10,  as  1000  and  100.     Thus 


TRIGONOMETRY 

1000  =  103 
100  =  102 


hence  1000  x  100  =  105  =  100,000, 

the  multiplication  being  performed  by  the  addition  of  exponents. 

Similarly,  if  384  =  102-58*53* 

and  25  =  101-39794+, 

384  may  be  multiplied  by  25  by  adding  the  exponents  of  102-584334-  and 
10i.39794+j  thus  obtaining  lO3-9822^,  and  then  getting  from  a  table  of  loga- 
rithms the  value  of  103-98227+,  viz.  9600. 

In  like  manner,  by  the  use  of  logarithms,  the  process  of 
dividing  one  number  by  another  is  converted  into  the  simpler 
process  of  subtracting  one  exponent,  or  log,  from  another ; 
the  process  of  involution  is  converted  into  the  simpler 
process  of  multiplication;  and  the  extraction  of  a  root  into 
the  simpler  process  of  division. 

The  saving  of  labor  effected  by  the  use  of  logarithms  can 
be  increased  by  committing  to  memory  the  logs  of  certain 

much  used  numbers  as  of  2,  3,  •••  9,  TT,  I/TT,  — ,  V  2,  J/3,  etc. 

TT 

Also  by  use  of  the  slide  rule,  the  practical  use  of  logarithms 
is  reduced  to  sliding  one  rod  along  another  and  reading  off 
the  number  corresponding  to  the  terminal  position  of  one 
end  of  a  rod.  If  the  teacher  can  find  time,  it  will  be  a  use- 
ful exercise  to  teach  the  class  the  use  of  the  slide  rule  in  con- 
nection with  the  study  of  this  chapter. 

3.    Systems  of  Logarithms.     Any  positive  number  except 
unity  may  be  made  the  base  of  a  system  of  logarithms. 
Two  principal  systems  are  in  use : 

1.  The  Common  (or  Decimal)  or  Briggsian  System,  in 
which  the  base  is  10.  This  system  is  used  almost  exclusively 
when  logarithms  are  employed  to  facilitate  numerical  compu- 
tations. 


LOGARITHMS  9 

•r 

2.  The  system  termed  Natural  or  Napierian,  in  which  the 
base  is  2.7182818+.  This  system  is  generally  used  in  alge- 
braic processes,  as  in  demonstrating  the  properties  of  algebraic 
expressions,  etc. 

EXERCISE  I 

1.  Give  the  value  of  each  of  the  following  :    Iog3  9,  Iog3  27,  Iog4  64, 
log4TV,  logg |,  logj^r,  log10TV,  logio-01,  Iog10.001. 

2.  Also  of  Iog2  32,  Iog2  gV,  !og2  yis >  }o^  8,  Iog8 16. 

3.  Simplify  Iog2  4  -f  Iog3  9  +  Iog10 .1  —  Iog3 1. 

4.  Write  out  the  value  of  each  power  of  2  up  to  220  (thus  21  =  2, 
22  =  4,  23  =  8,  etc.)  in  the  form  of  a  table. 

5.  By  means  of  this  table  multiply  32  by  8,  converting  the  multi- 
plication into  an  addition  of  exponents. 

6.  In  like  manner  convert  each  of  the  following  multiplications  into 
an  addition :  32  x  16 ;  64  x  32 ;  1024  x  16 ;  512  x  64. 

7.  Also  convert  each  of  the  following  divisions  into  a  subtraction  : 
1024-16;  512-64;  32768-8-1024. 

8.  Also  convert  each  of  the  following  involutions  into  a  multiplica- 
tion:  (32)3;  (64)2;  (32)4. 

9.  Also  convert  each  of  the  following  root  extractions  into  a  divi- 
sion: ->/64;  -S/1024;  A/4096. 

•  10.    Let  the  pupil  make  up  two  examples  like  those  in  Ex.  6 ;  in 
Ex.  8 ;  in  Ex.  9. 

11.  Let  the  pupil  construct  actable  of  powers  of  3  and  make  up 
similar  examples  concerning  it. 

COMMON   SYSTEM 

4.  Characteristic  and  Mantissa.  If  a  given  number,  as 
384,  be  not  an  exact  power  of  the  base,  its  logarithm,  as 
2.58433+,  consists  of  two  parts,  the  whole  number  called  the 
characteristic,  and  the  decimal  part  called  the  mantissa. 

To  obtain  a  rule  for  determining  the  characteristic  of  a 
given  number  (the  base  being  10),  we  have, 

10,000  =  104,  hence  log  10,000  =  4 ; 

1000  =  103,  hence  log  1000  =  3 ; 

100  =  102,  hence  log  100  =  2 ; 

10  =  101,  hence  log  10  =  1. 


10  TRIGONOMETRY 

f, 

Hence  any  number  between  1000  and  10,000  has  a  loga- 
rithm between  3  and  4  ;  that  is,  the  log  consists  of  3  and  a 
fraction.  But  every  integral  number  between  1000  and 
10,000  contains  four  digits.  Hence  every  integral  number 
containing/bwr  figures  has  3  for  a  characteristic. 

Similarly  every  number  between  100  and  1000,  and  there- 
fore containing  three  figures  to  the  left  of  the  decimal  point, 
has  2  for  a  characteristic  ;  every  number  between  10  and 
100  (that  is,  every  number  containing  two  integral  figures) 
has  1  for  a  characteristic  ;  and  every  number  between  1  and 
10  (that  is,  every  number  containing  one  integral  figure)  has 
0  for  a  characteristic. 

Hence,  the  characteristic  of  an  integral  or  mixed  number  is  one 
less  than  the  number  of  figures  to  the  left  of  the  decimal  point. 

5.   Characteristic  of  a  Decimal  Fraction. 
1  =  10°.  '  .-.  log  1  =  0  ; 
.1  =  ^  =  10-      .-.log  .!=-!; 

•  m===^-  •••'<*  •<*—* 


•••!<*  -°01  =  -3,  eta. 

Hence  the  logarithm  of  any  number  between  .1  and  1  (as 
of  .4  for  'instance)  will  lie  between  -  1  and  0  and  hence  will 
consist  of  --  1  plus  a  positive  fraction  ;  also  the  logarithm  of 
every  number  between  .01  and  .1  (as  of  .0372  for  instance) 
will  be  between  —2  and  —1,  and  hence  will  consist  of 
plus  a  positive  fraction  ;  and  so  on. 

Hence,  the  characteristic  of  a  decimal  fraction  is  negative, 
and  is  numerically  one  more  than  the  number  of  zeros  between 
the  decimal  point  and  the  first  significant  figure. 

There  are  two  ways  in  common  use  for  writing  the  char- 
acteristic of  a  decimal  fraction. 

Thus,  (1)  log  .0384=2.58433,  the  minus  sign  being  placed  over  the  char- 
acteristic 2,  to  show  that  it  alone  is  negative,  the  mantissa  being  positive. 


• 


LOGARITHMS  11 

Or  (2)  10  is  added  to  and  subtracted  from  the  log,  giving  ^ 

log  .0384  =  8.58433  -  10. 

In  practice  the  following  rule  is  used  for  determining  the 
characteristic  of  the  logarithm  of  a  decimal  fraction  : 

Take  one  more  than  the  number  of  zeros  between  the  decimal 
point  and  the  first  significant  figure.,  subtract  it  from  10,  and 
annex  —  10  after  the  mantissa. 

EXERCISE  2 

Give  the  characteristic  of  : 

1.  452.  6.    .08267.  11.    7. 

2.  16730.  7.    1.0042.  12.   6267.3. 

3.  767.5.  8.    7.92631.  13.   .000227. 

4.  64.56.  9.    .007.  14.   100.58. 

5.  9.22678.  10.    .0000625.  15.   23.7621. 

16.  How  many  figures  to  the  left  of  the  decimal  point  (or  how  many 
zeros  immediately  to  the  right)  are  there  in  a  number  the  characteristic 
of  whose  logarithm  is  3?  2?  5?  1?  0?  4?  8-10?  7-10?  9-10? 

17.  Can  you  make  up  a  rule  for  fixing  the  decimal  point  in  the 
number  which  corresponds  to  a  given  logarithm? 

6.  Mantissas  of  numbers  are  computed  by  methods,  usually 
algebraic,  which  lie  outside  the  scope  of  this  book.  After 
being  computed  the  mantissas  are  arranged  in  tables,  from 
which  they  are  taken  when  needed.  In  this  connection  it  is 
important  to  note  that 

The  position  of  the  decimal  point  in  a  number  affects  only 
the  characteristic,  .not  the  mantissa,  of  the  logarithm  of  the 
member. 

Thus,  if  log  6754  =  3.82956 


C.T'U 

log  67.54  =  log  ~  =  log  -  =  log  101-82956  =  1.82956. 

In  general  log  6754  =  3.82956 

log  675.4  =  2.82956 
log  67.54  =  1.82956 
log  6.754  =  0.82956 
log  0.6754  =  9.82956  -  10 
log  0.06754  =  8.82956  -  10,  etc. 


12  TRIGONOMETRY 

7.  Direct  Use  of  a  Table  of  Logarithms ;  that  is  given  a 
number,   to  find  its  logarithm.     For  methods  in  detail   see 
Introduction  to  Logarithmic  Tables  (Arts.  2-5  and  17). 

EXERCISE    3 

Using  five-place  tables  find  the  logarithm  of  each  of  the  following 
numbers : 

1.  7627.  10.  .00672.  19.  17.6287. 

2.  6720.  11.  .000007.  20.  42. 

3.  82.  12.  400000.  21.  .000001. 

4.  7862.  13.  14.6235.  22.  .0186789. 

5.  75.  14.  .00226725.  f23.  32679. 

6.  157.  15.  87.  24.  3267.9. 

7.  36278.  16.  .76.  125.  326.79. 

8.  67.222.  17.  .000125.  (26.  32.679. 

9.  3.3427.  18.  100.25.  V27.  3.2679. 

28.  Commit   to   memory  the   mantissa  for  each  of  the  following : 
2,  3,  5,  TT.     Then  write  at  sight  the  log  of  each  of  the  following,  200, 

3000,  50,  100  TT,  20,  .002, 30,  .0005,  -£-,  .3,  .2,  10  *,  20,000. 

100 

Using  four-place  tables,   find   the  logarithm   of  each  of  the   fol- 
lowing : 

29.  12.67.  36.  24.68.  43.  .000036775. 

30.  762.8.  37.  .11116.  44.  .0026382. 

31.  42.68.  38.  11.685.  45.  28966. 

32.  1.2267.  39.  .0012678.  46.  19.572. 

33.  .0263.  40.  965.3.  47.  .8625. 

34.  .0012678:  41.  1.4676.  48.  .0100267. 

35.  1.0026.  42.   1.7628.  49.   2.225. 
50.   Work  Ex.  28  for  four-place  tables. 

8.  Inverse  Use  of  a  Table  of  Logarithms;   that  is,  given 
a  logarithm,  to  find  the  number  corresponding  to  it  (called 
its    antilogarithm).     See   Introduction   to   the   Logarithmic 
Tables  (Arts.  6  and  17). 


LOGARITHMS  13 

EXERCISE   4 

Using  five-place  tables,  find  the  antilogarithm  of  each  of  the  follow- 
ing: ' 

1.  1.41863.   ;   ,,-  •     4.   7.68416.  7.   6.59068. 

2.  2.19756.  5.   9.22321-10.  8.   5.74706-10. 

3.  0.98349.  6.   6.42857-10.  9.   8.00400. 

10.  Find  log  of  2.34578.  15.  Find  antilog  of  3.21678. 

11.  Find  antilog  of  2.34578.  16.  Find  antilog  of  6.00371. 

12.  Find  log  of  1.03678.  17.  Find  log  of  6.00371. 

13.  Find  antilog  of  1.03678.  18.  Find  antilog  of  4.98672. 

14.  Find  log  of  3.21678.  19.  Find  log  of  4.98672. 

Find  the  number  corresponding  to  each  of  the  following  logarithms, 
using  four-place  tables. 

20.  1.4082.       23.   9.1546-10.       26.   8.0283-10.      29.   2.6575. 

21.  2.7332.       24.   2.0326.  27.    7.1170-10.       30.   4.3490-10. 

22.  3.2335.       25.   1.0135.  28.   5.0019-10.       31.   2.8177. 

32.  Find  antilog  of  2.3041.  35.   Find  antilog  of  0.4975. 

33.  Find  log  of  2.3041.  36.    Find  antilog  of  1.6924. 

34.  Find  log  of  0.4975.  37.   Find  log  of  1.6924. 

COMPUTATIONS   BY   USE  OF   LOGARITHMS 

9.    Properties  of  Logarithms  used  in  Numerical  Computations. 
It  is  shown  in  algebra  that 

a*  .  tf  =  ax+y'y  and  also  that  (ax)p  =  ap*. 
Using  these  properties  of  exponents,  it  can  be  shown  that 

1.  log  (mn)  =  log  m  4-  log  n.     3.    logmp   —P  log  m. 

fm\  PI —     log  m 

2.  log    f—  J  =  logm-logn.     4.    logVm=  — — 

For     771  =  10*.     .-.  logm  =  a?. 
n  =  10y.      .*.   logn  —  ?/. 
.-.  mn  =  10*+y  or  log  mn  =  x  +  y  =  logm  +  log  n.  (1) 

Also       = :      =  10*-",  or  log  ^  =  x  -  y  =  log  m  -  log  n.  (2) 


14  TRIGONOMETRY 

Also  mp  =  (WX)P  =  Wpx.     .-.  log  mp=px=p  -  log  m,  (3) 


and       Vm  =  Wp.     .-.  log  Vm  =    =  -.  (4) 

P         P 
Hence  : 

I.  To  multiply  numbers  : 

Add  their  logarithms  and  find  the  antilogarithm  of  the  sum. 
This  will  be  the  product  of  the  numbers. 

II.  To  divide  one  number  by  another  : 

Subtract  the  logarithm  of  the  divisor  from  the  logarithm 
of  the  dividend  and  obtain  the  antilogarithm  of  the  difference. 
This  will  be  the  quotient. 

III.  To  raise  a  number  to  a  required  power  : 

Midtiply  the  logarithm  of  the  number  by  the  index  of  the 
required  power  and  find  the  antilogarithm  of  the  product. 

IV.  To  extract  the  required  root  of  a  number  : 

Divide  the  logarithm  of  the  number  by  the  index  of  the 
required  root  and  find  the  antilogarithm  of  the  quotient. 

Ex.  1.  Multiply  561.75  by  .03286  by  the  use  of  loga- 
rithms. % 

log  (561.75  x  .03286)=  log  561.75  +  log  .03286 
log  561.75  =  2.74954 
log  .03^86  =  8.51667-10 

antilog  1.26621          =18.4591,    Product. 

The  following,  however,  is  the  arrangement  of  work  used 
by  many  practical  computers.  It  has  the  advantage  of  show- 
ing all  the  steps  in  a  complex  logarithmic  computation.  (See 
p.  12,  etc.) 

561.75  log  2.74954 
.03286  log  8.51667  -  10 
Answer  =  18.4591  log  1.26621 
Observe  that  "561.75  log  2.74954"  reads  "561.75,  its  log  is  2.74954,"  etc. 

Ex.  2.  Compute  the  amount  of  $  1  at  5  per  cent  com- 
pound interest  for  20  years. 


LOGARITHMS  15 

The  amount  of  $  1  at  5%  for  20  years  =  (1.05)20. 
1.05  log  0.02119  ;  20  log  0.42380 
Amount  =  2.65338  log  0.42380. 

If  the  student  will  compute  the  value  of  (1.05)20  by  con- 
tinued multiplication,  and  compare  the  labor  in  such  a  pro- 
cess with  that  involved  in  the  above  process,  he  will  have  a 
good  illustration  of  the  usefulness  of  logarithms. 

Ex.  3.     Extract  approximately  the  cube  root  of  532.768. 

532.768  log  2.72653  1  log  0.90884. 
Root  =  8.1066  log  0.90884. 

10.  Cologarithm.    In  operations  involving  division,  instead 
of  subtracting  the  logarithm  of  the  divisor,  it  is  usual  to  add 
its  cologarithm.     The  cologarithm  of  a  number  is  obtained 
by  subtracting  the  logarithm  of  the  number  from  10  — 10. 
Hence  adding  the  cologarithm  of  the  divisor  gives  the  same 
result  as  subtracting  its  logarithm.     The  use  of  the  cologa- 
rithm saves  figures,  and  gives  a  more  orderly  and  compact 
statement  of  the  work. 

;he  cologarithm  of  a  number  is  obtained  directly  from  a 
e  of  logarithms  by  the  following  rule : 
Subtract  each  figure  of  the  logarithm  of  the  given  number 
from   9    except    the  last  significant  figure,    which    subtract 
from  10. 

Ex.  1.     Find  the  colog  of  37.16. 

log  37.16  =  1.57008. 
Hence,  colog  37.16  =  8.42992  -  10. 

Ex.  2.  Divide  52678  by  37.16  by  the  use  of  the  cologa- 
rithm of  the  divisor. 

52678     log  4.72163 
37.16  log  1.57008  colog  8.42992  -  10. 
Quotient  =  1417.58  log  3.15155. 

11.  In  the  extraction  of  the  root  of  a  decimal  number  it  is 
best  to  add  to  and  subtract  from  the  logarithm  of  the  decimal 


16  TRIGONOMETRY 

number  such  a  multiple  of  10  that  the  last  term  of  the  quotient 
shall  be  10. 

Ex.    Extract  the  seventh  root  of  .0854329. 

.0854329  log  8.93162  -  10 

60  -60 

7)68.93162  -  70 

Root  =  .703667  log  9.84737  -  10 

12.  Computations  involving  Negative  Numbers.  In  com- 
puting, by  the  use  of  logarithms,  the  value  of  expressions 
containing  one  or  more  negative  factors,  first,  determine  the 
sign  of  the  result ;  second,  determine  the  magnitude  of  the 
result  by  treating  all  the  factors  as  if  they  were  positive  and 
using  logarithms. 

Ex.   Compute  ~        . 

/  y£) 

The  result  must  be  negative,  since  a  negative  number 
divided  by  a  positive  number  gives  a  negative  quotient. 
The  magnitude  of  the  result  is  determined  by  computing 

876 

EXERCISE    5 


the  value  of 


795 


Compute-by  mean's  of  five-place  logarithms  the  value  of 
each  of  the  following : 

1.  85  x  627.  5.  45  x  27.68  x  .0967  x  4.2678. 

2.  26.27  x  52.67.  ,    6.  (2.67)3. 

3.  8.25x25675.  ?  27.8675 

1768  18.678' 

'    211.6'  8.  (.5278)7. 

9.  -\/156.78.  Also,  if  you  can,  extract  the  cube  root  of  156.78  with- 
out the  use  of  logarithms.  About  how  much  more  work  in  this  process 
than  in  the  logarithmic  process  ?  Which  process  is  more  likely  to  be 
accurate,  the  long  or  the  short  one  ? 

10.    -\/.86785.     Also  extract  the  square  root  of  the  square  root  of 


LOGARITHMS  17 

.86785.     About  how  much  longer  is  this  process  than  the  logarithmic 
work  ? 

11.    \/-  76.526.         12.    -\/-. 00021.         13.    -fy -  .00062367  x  7.867. 

Find  the  compound  interest  on : 

14.  $  15375  for  20  years  at  6%.     Make  the  computation  without  the 
use  of  logs.     What  fraction  of  the  work  is  avoided  by  the  use  of  logs  ? 

15.  $  323.50  for  12  years  at  8%. 

16.  In  1623  the  Dutch  bought  Manhattan  Island  from  the  Indians 
for  $  24.     What  would  this  sum  amount  to  at  the  present  time,  if  it 
had  been  placed  on  interest   at  6%,  the  interest   to  be  compounded 
annually  ? 

17.  By  aid  of  the  logs  committed  to  memory  in  Ex.  28,  page  12, 

200     100  TT    300  x  500 

compute  each  01  the  following  :   ^=^;  ;   -  — 

oTo        oo    "  TT 

18.  Also  obtain  the  colog  of  43560  (the  number  of  square  feet  in  an 
acre)  and  use  it  to  find  the  area  in  acres  of  a  field  200  ft.  x  300  ft. ; 
one  300  ft.  x  500  ft. ;  one  1000  ft.  x  2000  ft. 

Using  four-place  logarithms,  compute  the  value  of  the  following: 


19.  1.2634  x  26.42. 

20.  .001467  x  96.8  x  47.37. 


J22.93 
\I6^T 


E 


556.85  x  .00016277  x  4.6.  .0016666 


(12.67)3.  '    .00042635' 


(3.176)7.          »  26.    V42.67  x  .10126  x  9.2. 

27.  A/.0000073. 

28.  Work  Exs.  17  and  18  by  the  four-place  tables. 

29.  Why  are  four-place  logarithmic  tables  sufficiently  accurate  for 
the  work  of  a  carpenter  or  land  surveyor  ? 

Find  the  compound  interest  on : 

30.  $  359.67  for  8  years  at  6%. 

31.  $  100  for  37  years  at  4  % . 

32.  $4962.75  for  16  years  at  5%.     Try  to  compute  this  without 
the  use  of  logs.     About  how  much  longer  is  the  process  without  logs  ? 
Which  process  is  more  likely  to  be  accurate  ? 

13.  Complex  Computations.  By  the  use  of  the  properties 
of  logarithms  demonstrated  in  Art.  9,  the  value  of  a  complex 
numerical  expression  may  be  computed. 


18  TRIGONOMETRY 


V215 
a~   '    eo-  by  the  use  of  logarithms. 
67  x  52    J 


2  =  *  log  6T^52  =  «*<*  215  +  C°log  67  +  colog  62>' 
Before  looking  up  the  logarithm  of  any  number  in  the  table,  it  is  im- 
portant to  make  a  scheme  or  outline  of  the  work,  leaving  blank  the 
places  which  are  to  be  filled  in  by  logs  taken  from  the  table.     Thus 
the  preliminary  outline  for  Ex.  1  would  be  as  follows : 

215  log 

67  log colog 

52  log colog 

2) 

Answer  = log 

After  looking  up  and  inserting  the  logarithms  and  completing  the 
computation,  the  work  will  appear  as  follows : 

215  log  2.33244 

67  log  1.82607  colog  8.17393  -  10 

52  log  1.71600  colog  8.28400  -  10 

2)18.79037  -  20 

Answer  =  .248422      log  9.39519  -  10 

One  advantage  of  the  above  method  of  tabulating  logarithmic  work 
is  that  without  essential  change  in  the  form  of  the  tabulating,  the  work 
may  be  presented  in  the  above  complete  form,  or  in  a  more  condensed 
form  (at  the  option  of  the  teacher),  as  by  omitting  the  logs  of  67  and 
52  and  giving  only  their  respective  cologs  in  the  tabulation. 

V2L8 .  A/.03678 , 
Ex.  2.    Compute  -  -  by  the  use  01  logarithms. 

.28756 

21.8  log  1.33846  -J  log  0.66923 

.03678  log  8.56561  -  10  1  log  9.52187  - 10  ' 
.28756  log  9.45873  -  10  colog  0.54127 
Answer  =  5.39975  log  0.73237 

14.  Exponential  Equations.  An  exponential  equation  is 
one  in  which  the  unknown  quantity  occurs  in  the  exponent 
of  some  term  or  factor,  as  ax  —  l}.  An  equation  of  this  kind 
can  often  be  solved  by  the  use  of  logarithms. 

Ex.   Find  the  value  of  x  in  the  equation  .3*  =  2. 


LOGARITHMS  19 


Taking  the  logarithm  of  each  member  of  the  equation, 

a;  log  .3  =  log  2. 
lo    2  0.30103  0.30103 


9.47712-  10  (X522 


*  = 


EXERCISE  6 

Using  five-place  tables,  compute  the  value  of  the  following : 

(Do  not  fail  to  make  an  outline  of  the  work  in  each  example  before  looking  up 
any  logarithms.) 

V2L82  x  V.0071725   .  /.59  x  2209 


.926Z8  ^  47  x  .3481 


2.    Y     ~     W  '  4"    V(.19678)2  -  (.072567)2. 

-V/.00231  X  V76l9~ 


.        /267.S5  x  7  x  .000925  x  468.765 

D. 


(21.67)2  x  .00096725  x^/567.256 

7.   Using  the  logarithms  committed  to  memory  in  Ex.  28,  Exer- 
cise 3,  compute  each  of  the  following: 


'300  x  500. 
j 


37 


8.  If  there  are  39.37  inches  in  a  meter,  convert  the  following  into 
feet:  500  meters;  7294  meters;  300  meters  (height  of  Eiffel  Tower). 
What  logs  used  in  the  first  of  these  computations  could  be  retained  and 
used  in  the  other  computations  ? 

Solve  for  x  : 

9.  6*  =  67.  11.   2.8*  =  .1967. 
10.   142*  +  3  =  2167.                                    12.    .85*  =  .  01978. 


*  If  the  teacher  prefers,  the  remainder  of  the  work  for  this  example  may  be 
arranged  as  follows  : 

log  x  +  log  (log  .3)  =  log  (log  2). 
/.  log  x  =  I  •  log  2  -  1  •  log  .3. 

2  log  0.30103  1  •  log  9.47861  -  10. 
.3  log  9.  47712  —  10  (or  —  ,.52288)  1  •  log  (-)  9.71840  —  10  colog  0.28160. 

x  =  -  .5757+  log  9.76021  -  10. 


20  TRIGONOMETRY 

13.  Find   the  side  of   a  square  whose  area  is  equal  to  that  of  a 
parallelogram  whose  base  is  22.678  and  whose  altitude  is  17.375. 

14.  Find  the  side  of  a  square  whose  area  is  equal  to  that  of  a 
circle  whose  radius  is  13.56. 

15.  Calculate  the  value  of  K  in  the  equation, 


/f  =  •     s(s  -  a)(s  -  b)(s-  c), 
wheni9_<l±A±_^  and  a  =  17.6,    6=21.675,    c  =  26.427. 

16.  Calculate  the  value  of  6  in  the  equation,  b  =  Va2  —  c2,  when 
a  =  .17623  and  c=  .12673.    (Use  6=  V(a  +  c)(a  —  c),  etc.) 

17.  Find  the  volume  of  a  sphere  whose  radius  is  14.7,  if  V=  f  irR3 
and  TT  =  3.1416. 

la    Given  t  =  8,    a  =  32.17,    find  s,  if  s  =  J-  at2. 

19.   Given  s  =  a  +  b  +  c  and  a  =  .1732,    b  =  .14326,   c  =  .2242,    find 


hj  if  h  =  -  Vs(s  —  d)(s  —  b)(s  —  c). 
c 

20.  Given  #  =  14.16  and  TT  =  -2f ,  find  5,  if  5  =  4  7r#2. 

21.  Given  *  =  -^  and  D  =  23.8,  find  F,  when  V  =  i  TrZ)3. 

22.  In   how   many   years   will    $1   at  compound   interest    at  5  % 
amount  to  $25? 


Using  four-place  tables, 

compute  the  value  of  the  following  : 

23. 
24. 

*/      529 

1(5.78    |i 

O  O'T 

\67  X  51.8 

25                A/ 

6.78 

/  .3756  x 

.265 

26.     V(125) 

(67)2. 

\   .227  x  . 

27. 
28. 

29. 

1678 
47.326 

/    55400  X  8 

.10021 

V  123456  x  .007 

-^.216 

7  ^     /21.67         1  .16765  _ 

\32.77      V  1.76364 

UN 

'1^673  (26.72)2   1  . 

(36.27)^  X  .01267 
Solve  for  x : 

30.  2*  =-19.  32.   19.383*  =  81672. 

31,  42jc-3  ,=  n«+i,  33.   .17'  =  .4782, 


LOGARITHMS  21 

34.  Find   the  side   of   a  square  whose   area  is  equal  to  that  of  a 
rectangle  whose  base  is  17.628  and  whose  altitude  is  8.263. 

35.  Find   the  volume  of  a  sphere  whose   radius   is   1.1124,  using 


36.  Given  t  =  12  and  g  =  32.17,  find  s,  if  s  = 

37.  Work  Exs.  16-19  above  by  the  use  of  four-place  tables 

38.  Work  Exs.  7  and  8  above  by  the  use  of  four-place  tables. 

GENERAL   PROPERTIES   OP   SYSTEMS   OF   LOGARITHMS 

15.  The   logarithm   of   unity  in   any  system  of  logarithms 
is  zero. 

For,  if  a  be  the  base, 

1  =  a°.     .-.  log,  1  =  0. 

16.  The   logarithm   of   the    base    in   any   system   of  loga- 
rithms is  unity. 

For  a  =  a1.     .-.  log,,  a  =  1. 

17.  The   logarithm   of   zero   in   any  system  whose  base  is 
greater  than  unity  is  negative  infinity  ;  that  is,  as  the  number 
approaches  0,  the  logarithm  approaches  negative  infinity. 

«or,   since   «>1,  0  =  —  =  ^  =  or00  .     .'.  log  0  •  =  -  oo. 
ut   in   any  system  whose   base  is  less  than   unity,  the 
logarithm  of  zero   is  positive  infinity. 

For,  since        a  <  1,  0  =  a00.     /.  loga  0  =  oo. 

18.  Logarithm  of  a  Product,  Quotient,  Power,  and  Root  in 

any  system. 

If  a  be  taken  as  the  base,  and  m  and  n  be  any  two 
numbers,  it  can  be  shown  in  a  manner  similar  to  that  used 
in  Art.  9  that 

1.  loga  mn  =  loga  m  +  loga  n. 

wi/ 

2.  loga  —    =  log«  m  —  loga  n.      [Let  the  pupil  supply  the 


3. 


proof.     See  Art.  9  ;  use 
-j.tog.rn.  Jfa 


22  TRIGONOMETRY 

19.  Changing  the  Base  of  a  System  of  Logarithms.  Given 
the  logarithm  of  a  given  number,  r,  to  a  base  a,  to  find  the 
logarithm  of  r  to  another  base  k,  we  use  the  following 
formula:  } 


For,  let  logj.  r  =  x. 

Then  #"      =r    ........     (1) 

by  definition  of  a  logarithm. 

Take  the  logarithm  of  each  member  of  (1)  to  base  a, 
then  x  loga  k  =  loga  r. 

Hence,  x  =  1  °a    ? 

loga& 

lo     r 


or 


It  follows  as  a  special  case  that  if  r  =  a, 

log*  a  = -,  or  log*  a-loga  k  =  1. 


Ex.   Find  the  logarithm  of  .7  to  the  base  5. 
By  the  formula  just  proved, 

9.84510-10 


• 


Iog10  5  0.69897 


EXERCISE  7 

In  working  the  first  twelve  examples  in  the  following  exercise  use 
four-place  tables  in  solving  the  even-numbered  examples,  and  five-place 
tables  in  solving  the  odd-numbered  examples. 

Find  the  value  of : 

1.  Iog5  60.  5.  log^VB.  9.  Iog2 .7261. 

2.  Iog69.3.  6.  logsolS.  10.  log^  -08275. 

3.  Iog3.726.2.  7.  logj.8 .17362.  11.  logL2 .9267. 

4.  Iog4.93.  8.  log.8.2631.  12.  Iog7  V3.1416. 


LOGARITHMS  23 

Find  without  the  use  of  tables : 

13.  Iog327.  15.   lo'g9JT.  17.   logo  .125. 

14.  Iog232.  16.    logj_8.  18.    Iog2.0625. 

19.  Find  the  base  of  the  system  of  logarithms  in  which  the  log  of 
16  =  4. 

20.  If  the  log  of  27  =  f ,  find  the  base. 

21.  If  i  =  the  log  of  5,  find  the  base. 

22.  Given  the  log  of  5^  =  —  f ,  find  the  base. 

23.  If  the  log  of  64  =  1.2,  find  the  base. 

24.  In  how  many  years  will  a  sum  of  money  double  itself  at  4  % 
compound  interest?  at  6  %  ? 

25.  If  $1520  amounts  to  $10,701.46  in  40  years  at  compound  inter- 
est, what  is  the  rate  per  cent  ? 

26.  Who  invented  logarithms,  and  when  (see  p.  169)?     Find  out  all 
you  can  about  this  man  and  the  way  in  which  he  invented  logarithms. 

27.  What  nation  first  divided  the  circle  into  360  degrees,  and  one 
degree  into  60  minutes  ? 


• 


CHAPTER    II 
DEFINITIONS.     TRIGONOMETRIC    FUNCTIONS 

20.  Source  of  New  Power.  Illustrations.  A  spring  of 
water  is  situated  at  the  point  A  and  a  house  at  B.  It  is 
desired  to  find  the  length  of  a  pipe  needed  to  connect  B 
with  A,  A  and  B  being  separated  by  a  swamp.  How  can 
the  length  of  the  pipe  be  determined  without  going  through 
the  swamp? 


32.0  yd. 
FIG.  1. 


510  yd. 
FIG.  2. 


FIG.  3. 


If  the  swamp  is  situated  as  in  Fig.  1,  so  that  a  point  C  can 
be  taken  where  CA  and  CB  form  a  right  angle,  then  CA 
and  CB  can  be  measured  and  the  length  of  AB  computed  by 
the  methods  of  plane  geometry.  Let  the  pupil  compute 
AB  of  Fig.  1. 

But  if  the  swamp  is  situated  as  in  Fig.  2,  the  above  method 
of  computing  AB  cannot  be  followed.  However,  if  we  take 
a  convenient  point  C  in  Fig.  2  and  measure  the  lines  A  C, 
CB,  and  the  Z  (7,  the  distance  AB  can  be  computed  provided 
we  have  a  table  giving  the  ratios  of  the  sides  of  all  possible 
right  triangles.  Thus  from  this  table  we  form  the  triangle 
given  (on  enlarged  scale)  in  Fig.  3.  Then  by  the  properties  of 
similar  triangles  we  have  the  proportion  10 : 5.2  =  420  yd. :  AD. 

24 


TRIGONOMETRIC   FUNCTIONS  25 

From  this  proportion  AD  is  obtained ;  afterward  AB  may 
be  computed  from  the  right  triangle  ADB  by  geometry. 

Hence  the  source  of  new  power  in  trigonometry  is  a  set 
of  tables  giving  the  ratio  of  each  pair  of  sides  in  all  possible 
right  triangles. 

By  the  aid  of  such  tables  it  will  be  found  that  we  are  able 
to  find  the  unknown  parts  of  many  tri- 
angles which  cannot  be  solved  by  ordinary 
geometry.     Thus  it  will  be  found  that  if 
one  side  AB  (Fig.  4)  and  any  two  angles 
(as  A  and  B)  of  a  triangle  be  known,  the 
other  sides  (AC  and   CR]  may  be  com- 
puted.    By  this  method,  for  instance,  the  FlG-  4- 
distance  from  the  earth  to  the  moon  is  computed.     (For  other 
illustrations  of  the   new  power  given  by  trigonometry  see 
Chapter  VII.) 

21.  Trigonometry,  as   first  considered,  is   that  branch  of 
mathematics   which   determines  the  remaining   parts   of   a 

t 

triangle  from  certain  given  parts. 

PThus  it  will  be  found  that  if  any  three  parts  of  a  triangle  are  given, 
provided  one  of  them  is  a  side,  the  remaining  parts  maybe  determined. 

Later  the  word  trigonometry  comes  to  have  a  more  ex- 
tended meaning  so  as  to  cover  the  theory  of  the  functions 
of  angles  in  general  wherever  these  angles  may  be  found. 
Hence  it  comes  to  include  much  of  the  theory  of  wave  motion 
and  therefore  of  particular  cases  of  wave  motion,  as  of  sound, 
light,  and  electricity.  It  also  becomes  largely  algebraic  in 
nature. 

Plane  Trigonometry  treats  of  plane  triangles. 

See  if  you  can  find  the  derivation  of  the  word  trigonometry. 

22.  Trigonometric  Functions  of  an  Acute  Angle.      The  fun- 
damental tools  or  instruments  used  in  trigonometry  are  the 
functions  of  an  angle  now  to  be  described  and  defined. 


26 


TRIGONOMETRY 


From  any  point  B  in  one  side  of  an  acute  angle  BAC 
let  fall  a  perpendicular  BC  to  the  other  side,  forming  the 
right  triangle  ABC. 


FIG.  5. 


6 
FIG.  6. 


T)  ri 

Then  the  ratio  — -  is  termed  the  sine  of  the  angle  A. 

Similarly, 

AC  AC  AB 

cosine  A  =  — — ,  cotangent  A  =  -^^,  cosecant  A  =  - 


tangent  A  =  — — ,  secant  J..=  -  — ,  versed  sine  A  =  l-    —  , 

coversed  sine  A  —  \ 

or,  in  general,  in  a  right  triangle  : 

The  sine  o/  an  acwte  angle  is  the  ratio  of  the  opposite  Mg 
to  the  hypotenuse. 

The  cosine  is  the  ratio  of  the  adjacent  leg  to  the  hypotenuse. 

The  tangent  is  the  ratio  of  the  opposite  leg  to  the  adjacent 
kg. 

The  cotangent  is  the  ratio  of  the  adjacent  leg  to  the  opposite 
kg. 

The  secant  is  the  ratio  of  the  hypotenuse  to  the  adjacent  leg. 

The  cosecant  is  the  ratio  of  the  hypotenuse  to  the  opposite  kg. 

The  versed  sine  is  1  minus  the  cosine. 

The  coversed  sine  is  I  minus  the  sine. 

These  eight  ratios  are  called  the  trigonometric  ratios,  or 
the  trigonometric  functions. 

The  versed  sine  and  the  coversed  sine  are  used  so  little  in 


TRIGONOMETRIC  FUNCTIONS  27 

elementary  work  that  we  confine  our  attention  mainly  to 
the  other  six  functions.  Hence  when  we  speak  of  the  "  six 
functions  "  we  mean  the  first  six  trigonometric  functions  as 
given  above. 

The  abbreviations  sin,  cos,  tan,  cot,  sec,  esc,  vers,  covers, 
are  ordinarily  used  for  the  eight  functions. 

The  cosine,  cotangent,  cosecant,  and  coversed  sine  are 
termed  the  co-functions  of  the  sine,  tangent,  secant,  and 
versed  sine  respectively. 

In  the  above  triangle  (Fig.  6),  denoting  the  side  AB  by 
c,  AC  by  &,  and  BC\>y  a,  we  have 


sm^i  =  -  aj/j^r  sec^4  =  - 

c  6 

b  rf  *  **                         c 

cos  A  =  -  esc  A  =  - 

c  a 

tan  A  =  -  vers  A  =  1 

b  ,                                     c 

-  covers^  =  l-- 


^ioailarly 


i 


a  -p,      c 

esc  B  =  — 

c  b 

tan  B—-  vers  J5  =  1  — 

a  c 

cotJ5  =  -  covers  B  =  l-b 

b  c 

Or  using  abbreviations, 

sin  of  either  acute  Z  =  •  1   PP*     cot  of  either  acute  Z  = 1 

hyp.  -L  opp. 

cos  of  either  acute  Z  = 1    sec  of  either  acute  Z  =  — 2Ei- 

hyp.'  J-- 


I  Vi     n 

tan  of  either  acute  Z  =  — EC,    esc  of  either  acute  Z  =  — 2£^- 

±adj.  -Lopp. 


28 


TRIGONOMETRY 


The  method  of  indicating  a  power  of  a  trigonometric 
function  is  shown  by  the  following  example:  for  "  the  square 
of  the  sine  of  the  angle  A"  that  is,  for  "  (sin^.)2,"  we  write 
"sin2^.."  How  then  would  "the  cube  of  cos^L"  be 
written?  "The  nth  power  of  tanJ.?" 

In  this  book  unless  the  contrary  is  stated,  in  the  right  triangle  ABC, 
the  letter  C  is  supposed  to  be  placed  at  the  vertex  of  the  right  angle. 

23.  Utility  of  the  Trigonometrical  Ratios.     It  will  be  found 
that  the  numerical  value  of  the  above  trigonometrical  ratios 
for   every   angle   from    0°   to  90°   may   be   computed   and 
arranged  in  tables  whence  they  may  be  taken  and  used  when 
needed.     These  numerical  values  are  used  by  what  is  vir- 
tually the  geometrical  principle  of  similar  triangles  in  solving 
triangles.     Later,  however,  they  become  units  and  elements 
which  can  be  variously  grouped  and  used  in  many  kinds  of 
algebraic  processes. 

24.  The  value   of   a  trigonometric   function    of   an  anorle 

depends  only  on  the  size  of  the  <u^^^ 
not  on  the  length  of  the  lines  cho«^H 
form  the  ratios. 

Thus,  by  similar  triangles  (in  Fig. 7), 


C'  C    C'J 
FIG.  7. 


=  ±^L  =  ±   -     etc 


AB'      AB     AB"' 


25.  Given  two  sides  of  a  right  triangle,  to  compute  the 
trigonometric  functions  for  both  acute  angles  of  the  triangle. 

Ex.  If  in  a  right  triangle  a  =  3,  and  6=4,  find  c  and  the 
trigonometric  ratios  of  each  acute  angle. 

The  hypotenuse  c  =  V32  +  42  =  V25  =  5 
Hence  sin  A  =  f     sin  B  =  f 

cos  A  =  f     cos  B  =  f 

tan^4.  =  |-    ,tanB  =  f 
etc.  etc. 


A 
FIG.  8. 


TRIGONOMETRIC   FUNCTIONS  29 

In  studying  trigonometry  (and  indeed  in  all  mathematical  work) 
the  pupil  should  make  the  capital  letter  a  in  the  printed  form  A  and 
not  in  the  script  form  Ct,*  In  other  words,  he  should  make  the  small 
and  capital  letters  as  unlike  as  possible,  and  hence  make  them  unlike  in 
shape  as  well  as  in  size.  The  reason  for  this  is  that  the  small  and  capi- 
tal letters  have  entirely  different  meanings ;  and  if  as  written  by  the 
pupil  they  have  the  same  shape,  the  pupil  is  continually  mistaking  the 
small  letter  for  the  large,  and  vice  versa.  Similarly  the  capital  letter 

c  should  always  be  written  in  the  form  ^&  and  not  (7. 

EXERCISE  8 
Y 

1.  Write  the  functions  of  the  acute  angle  B  (Fig.  6)  in  terms  of 

a,  6,  c.     (Let  the  teacher  invert  the  triangle  in  various  ways.) 

2.  Construct  a  right   triangle  in  which  a  =  8,  6  =  6,  c  =  10,  and 
write  out  the  functions  of* A  in  this  triangle ;  also  of  B. 

Determine  the  value  of  the  functions  of  A  in  the  rt.  A  ABC,  whose 
sides  are  a,  6,  c,  if : 

^      3.^  =  6,     6  =  8.  6.    a  =  39,    6  =  80. 

4.    a  =  8,     6  =  15.  7.    a  =  .09,    c  =  .41. 

».    a  =  12,    c  =  13.  8.    6  =  12,    .c  =  16.9. 

|9.    Find  the  value  of  the  functions  of  B  in  Exs.  3-8. 

I 

In  Ex.  2  find  the  value  of 

sin  A  tan  A.  (4)  1  -(-  tan2  A:  ^  ,      ^  _  sin^i 


(2)  sin2  A  +  cos2  A.         (5)  sec2  A  -  tan2  A.  cos  A 

(3)  sin  .4  esc  A.  (6)  tan  JL  cot  A.  (8)  cos  .4  sec  A 

By  the  use  of  squared  paper  construct  the  angle  whose 

11.  Tangent  =  f.  16.    sine         =|. 

12.  Tangent  =  1.  17.   cosine     •=£. 

13.  Tangent  =  1.  18.   secant     =  V3. 

14.  Tangent  =  4.  19.    cosecant  =  5. 

15.  Tangent  =  V3. 

20.  Construct  with  a  protractor  an  angle  of  23°.  Then  construct  a 
right  triangle  with  sides  of  convenient  length  having  23°  for  one  of  its 
angles.  Measure  the  sides  of  this  right  triangle  and  hence  find  sin  23°. 
Compare  this  value  with  the  value  of  sin  23°  given  in  Table  V.  Deter- 
mine and  test  cos  23°  and  tan  23°  in  the  same  way. 


30  TRIGONOMETRY 

\"* 

21.  Treat  37°  in  the  same  way ;  also  52°.  , 

22.  On  Fig.  2  (p.  24)  compute  the  numerical  value  of  AD\  then 
of  CD  and  Z>£;  then  of  AB. 

23.  On  Fig.  3,  what  is  the  value  of  sin  A'  ? 

24.  On  Fig.  6,  if  AB  =  125,  Z£  =  27°,  and  sin 27°  =.454,  compute 
AC. 

25.  Can  you  suggest  some  practical  problem  similar  to  that  given 
in  Art.  20,  which  could  be  solved  by  trigonometry  and  not  by  geom- 
etry ?     What   is   the    source   of    new   power   in   trigonometry   which 
enables  us  to  do  this  ? 

26.  If  by  the  methods  of  trigonometry  we  are  able  to  solve  any 
triangle  in  which  one  side  and  any  two  angles  are  given,  suggest  some 
practical  problem  which  could  be  solved  by  this  means  (and  not  by 
geometry). 

• 
In  a  rt.  A.  given : 

27.  a =  Vp*  4-  g2,  b  =  V2pg,  find  sin  A  and  cos  A. 

28.  a  =  2  mn,  c  =  m2  +  n2,  determine  sin  A,  sec  A,  and  tan  A. 

29.  b  =  2pq,  c=p*  +  q2,  find  tan  A,  sin  A,  esc  A. 

30.  a  —  Vm2  +  mn,  b  =  Vwn  +  n2,  find  all  the  functions  of  B. 

31.  If  a  =  2  Vmn  and  c  =  m  -f  n,  find  all  the  functions  of  B. 

32.  If  a  =  60  and  c  =  61,  find  sec  A,  tan  B,  cot  B,  sin  A. 

33.  If  b  =  2.64  and  c  =  2.65,  find  the  functions  of  B.  

34.  If  a  =  2  b,  find  the  functions  of  A. 

35.  If  b  —  |  c,  find  the  functions  of  A. 

36.  If  a  +  b  =  |  c,  find  the  functions  of  B. 

37.  If  a  —  b  =  ^  c,  find  the  functions  of  A. 

38.  Find  the  functions  of  B,  if  a  =  4  d  and  6  =  3  d 

By  use  of  squared  paper  construct  a  rt.  A,  given : 

39.  c  =  4  and  tan  A  =  %. 

40.  b  =  3  and  sin^=|.. 

41.  Find  b  if  cos  A  =  .36  and  c  =  4.5. 

42.  On  Fig.  8,  sin  A  =  what  ?    cos  B  =  what  ?  Does  sin  ^4  =  cos  B  ? 
In  like  manner,  show  that  cos  A  =  sin  B}  tan  ^4  =  cot  B,  cot  ^4  =  tan  B, 
sec  u4  =  esc  JB,  esc  ^L  =  sec  B. 

43.  Show  the  same  on  Fig.  6. 


TRIGONOMETRIC   FUNCTIONS  31 

44.  In  Fig.  6,  since  c  is  the  hypotenuse,  it  is  evident  that  it  is 

greater  than  either  leg.     Hence  sin  A,  or  -,  is  always  less  than  1. 

c 

What  other  function  of  A  is  always  less  than  1?  Which  functions 
of  A  are  always  greater  than  1  ?  Which  may  be  either  greater  or 
less  than  1  ? 

45.  Which   of    the    six   functions   are    always    proper    fractions? 
improper   fractions?    may  be   either   proper   or   improper   fractions? 
Verify  this  on  Fig.   8. 

46.  If  A  is   any  acute   angle,  is  it  correct  to  say  that  sec^l  is 
always  greater  than  sin  A  ?     Why  ? 

47.  The  values  of  which  of  the  six  functions  of  A  (on  Fig.  6)  have 
c  for  a  denominator  ?    a  ?    b  ? 

48.  How  many  of    the   above   examples   can   you   work   at   sight 
(i.e.  for  how  many  can  you  give  results  without  the  use  of  pencil  and 
paper)? 

26.    Functions   of    the   Complement  of    an  Angle.     From 

F«.  6  (page  26). 

sin  A  =  -  ;  also  cos  B  =  - . 
c  c 

sin  A  =  cos  B, 

sin  A  =  cos  (90°  -  A\  since  B  =  90°  -  A. 

Let  the  pupil  show  in  like  manner  that 

cos  A  =  sin  B  =  sin  (90°  -  A), 

tan  A  =  cot  B  =  cot  (90°  -  A)9 

and  sec  A  =  esc  B  =  esc  t(90°  -  A). 

Hence,  in  general, 

Any  trigonometric  function  of  an  angle  is  equal  to  the  co- 
function  of  the  complement  of  the  angle. 

By  the  use  of  this  property, 

Any  trigonometric  function  of  an  angle  between  45°  and  90° 
can  be  reduced  to  the  function  of  an  angle  between  0°  and  45°. 

Thus,  sin  88°  10'  =  cos  1°  50'. 


32 


TRIGONOMETRY 


5.  csc  21°  24'  30". 

6.  sec8'4°16'. 

7.  sin  89°  59'. 

8.  cos  1°  18'. 


EXERCISE  9 

Express  each  of  the  following  trigonometric  functions  as  a  function 
of  the  complementary  angle  : 

1.  sin  60°. 

2.  cos  15°. 

3.  tan  65°  24'. 

4.  cot  55°  36'. 

9.    Given  tan  60°  =  V3,  find  cot  30°. 

10.  Given  sin  30°  =  |,  find  cos  60°. 

11.  Given  cos  A  =  -,  find  sin  (90°  —  A). 

12.  Given  sin  A  =  p,  find  cos  (90°  —  A). 

13.  How  many  of  the  examples  in  this  exercise  can  you  work  at 
sight  ? 

RELATIONS   OF   TRIGONOMETRIC   FUNCTIONS   OF   AN  ANGLE 

27.    Three    pairs    of    reciprocals  exist  among  the  trigoB- 
metric  functions  of  an  acute  angle,  viz. 

sin   and  csc 

cos  and  sec 

tan  and  cot 

For 


a 


b 

FIG.  9. 


sin  A  x  csc  A  =  1 . 
cos  A  x  sec  A  =  1. 
tan  A  x  cot  A  =  1 . 


28.    Four  equations  connect  the  trigonometric  functions  of 
an  acute  angle  in  important  ways. 

For,  from  Fig.  9, 

a*  +  b*  =  <?.  ........     (1) 


TRIGONOMETRIC   FUNCTIONS  33 

Dividing  (1)  by  c2, 


that  is,  sin2  A  +  cos2  ^4  =  1. 

Dividing  (1)  by  W, 

fr-'r  -  (!)'-=(§)< 

that  is,  tan2  A  +  l  =  sec2  A  . 

Let  the  student  prove  in  like  manner  that 

cot2  A  +  1  =  esc2  A. 
Also  from  Fig.  9. 


s,  tan  A  = 


_^_. 
6      c     c 


cos^l 


Hence  nine  (or  more)  formulas  give  important  values 
e  trigonometric  functions.  For  from  the  results  of 
27  and  28  we  readily  obtain,  for  instance, 

fn^=Vl_cos^.  cot^=c^4 


,—  sin  A 

cos  A  .  —  V  1  —  sin2  A. 

«  sec  A-  =  ~       ~7' 

cos  A 


cos  A  -, 

i  esc  A  .=         —. 

tan  A  =  -  sin  A 

cotA  versA=  I  -cos  A. 

covers  A  =  1  —  sin  A. 

30.    One  trigonometric  function  of  an  angle  being  given,  the 
other  functions  may  be  found  in  either  of  two  ways.  - 

ALGEBRAIC  METHOD.     By  use  of  the  formulas  of  Art.  29 
and  equations  of  Art.  28. 


34  TRIGONOMETRY 

Ex.  1.     If  sin  A  =  f  ,  find  the  other  trigonometric  func- 
tions of  A. 

cos  A 


esc  A  = 

sm  ^ 

vers  -4  =  1  —  cos  ^4.  =  1  —  ^  V5. 
covers  .4  =  1  —  sin  ^L  =  1  —  -|  =  J. 

Ex.  2.     If  tan  x  =  2,  find  the  other  functions  of  x. 

sec2  x  =  1  +  tan2  x.     (Art.  28.) 
.'.  sec2  x  =  1  +  4  =  5. 
sec  x  =  V5. 


sec  x      V5      5 


sin  a;  =  Vl  —  cos2  x  =  VI  —  -J-  =  V|  =  f  V5,  etc. 

GEOMETRIC   METHOD.      This    consists   of   construct 
right  triangle  by  use  of  the  given  function  and  derivin 
required  functions  from  the  right  triangle. 

Ex.    3.      Given  sin  .A  =  f,  obtain  the  other  trigonometric 
functions  of  A.  by  use  of  the  right  triangle. 

Construct  a  right  triangle  whose  hypotenuse  is  3  and  altitude  is  2, 


Then     AC  =  V32  -  22  =  V9  -  4  =  V5. 
Then  from  the  figure  by  the  definitions  of  the 
trigonometric  ratios 

2 


FIG.  10.  covers  A  —  \—     = 


TRIGONOMETRIC  FUNCTIONS         t  35 

As  the  sides  of  a  right  triangle  are  all  positive  in  sign,  in  studying 
the  trigonometry  of  the  right  triangle  we  neglect  the  ±  sign  usually 
placed  before  a  square  root  radical  sign,  and  take  any  square  root 
radical  as  normally  plus.  When  we  come  to  study  angles  in  general, 
as  in  Chapters  IV  and  V,  it  will  be  necessary  carefully  to  consider 
whether  the  sign  before  a  given  radical  sign  is  to  be  taken  as  +  or  — 
(see  Art.  61). 


EXERCISE  10 

by  means  of  the  formulas  the  values  of  the  other  functions  of 

A,  given : 

s 
1.    sin  A  —  if.  5.    cot  A  —  m.  9.   tan  ^4  =  0. 

2.v  tan  A  =  -1/.  6.   csc  A  =  V5.  10.    sin  A  =  l. 

3.  *  sec  A  =  -^g1-.  7.    sin  .4  =  0.  11.   sec  A  =  GO  . 

4.  cos^4  =  f.  8.   cos  .4  =  0.  12.   sin#  =  5p. 

Find  by  geometric  methods  (squared  paper  may  be  used  to  advantage 
in  constructing  diagrams)  the  other  functions  of  A  (or  a?),  given : 

A  13.    tan  A  =  f .  16.    cot  A  =  f .  19.   tan  .4  =  m. 

14.    cos  ^4  =  3%.  17.    sin  A  =  \.  20.    sin  .4  =  i  V2. 

.   csc  A  =  il.  18.    sec  A  =  4.  21.   cos  a;  =  1. 

1  o 

d  by  both  methods  the  other  functions  of  the  angle  named  when  : 
.   cscJ.  =  44.  27.    cos  A  =  |. 


2  run       / 

=  —  -  •  7 

m-?i    f 


4 
23.   tan  A  =  —  -  •  7  28.    sec  A  = 


_  V6-V2- 

24.  cotJ.  =  V'2  +  l. 

29.    cos^4  =  /i". 

25.  sin  ^4  =  1. 

26.  tan  22|°  =  V2  -  1.  30.    cot!5°  =  2+V3. 

Express  each  of  the  other  trigonometric  functions  of  A  in  terms  of: 

31.  sin  A.  38.  Given  sin  A  =  f  ,  find  cot  A. 

32.  cos  A.  39.  Given  cos  A  =  f  |,  find  csc  A. 

33.  tan  A  40.  Given  tan  A  =  V3,  find  sin  A 

34.  cot  A  41.  Given  csc  A  =  f  ,  find  sec  A. 

35.  sec  A.  42.  Given  sec  A  =  -2^5,  find  cot  A. 

36.  csc  A  43.  Given  cot  A  =  V2  —  1,  find  cos  A 

37.  vers  A.  44.  Given  tan  A  =  V6,  find  csc  A. 


36  TRIGONOMETRY 

45.  Transform  the  expression  sin2  A  -h  cos  A  so  that  the  only  trigo- 
nometric function  contained  in  it  shall  be  cos  A. 

46.  Transform  (1  +  tan2  A)  sec  A  so  that 'it  shall  contain  only  cos  A. 

47.  Transform  (tan  A  +  cot  A)  sec  A  cos  A  so  that  it  shall  contain 
only  sin  A  and  cos  A. 

48.  Transform  the  equation  cos2  x  —  sin2  x  =  'simjs  'so  that  it  shall 
contain  only  sin  x. 

49.  Transform  tan  x  =  2  +  cot  x  so  that  it  shall  contain  only  tan  x. 

50.  Which  of  the  six  functions  are  always  less  than  1  ?     Which  are 
always  greater  than  1  ?     Which  may  be  either  greater  or  less  than  1  ? 
How  can  you  use  this  principle  in  testing  the  accuracy  of  examples  like 
Exs.  1-30  of  this  Exercise  ? 

51.  How  many  of  the  above  examples  can  you  work  at  sight  ? 

31.    Trigonometric  Identities. 

As  stated  in  algebra,  an  identity  is  an  equality  which  is  true  for  all 
values  of  the  unknown  quantity  (or  quantities)  contained  in  it. 

Thus  (x  4-  2)  (x  —  2)  =  x2  —  4  is  an  identity,  since  it  is  true  for  9 
values  of  x,  as  for  a?  =  0,  1,  2,  3,  ••-,  or  —  1,  —2,  etc. 

An  equation  proper  (or  a  conditional  equation)  is  an  equality 
is  true  only  for  a  certain  special  value  (or  values)  of  the  un 
quantity  (or  quantities). 

Thus  x2—  x  =  2x  —  2  is  true  only  when  x  =  1  or  2,  and  hence 
equation  proper,  or  conditional  equation.  - 

The  equality  mark  used  in  equations  is  =,  and  that  used  in  identities 
is  =.  However,  in  elementary  mathematics  it  is  customary  to  use  the 
mark  =  for  both  equations  and  identities  and  let  the  context  decide 
whether  we  are  dealing  with  an  identity  or  an  equation. 

Similarly  in  geometry  the  word  "  circle  "  is  sometimes  used  to  denote 
an  area  and  sometimes  a  line  (the  circumference),  the  context  deciding 
in  each  case  what  is  meant.  So  8"  may  mean  either  8  inches  or  8 
seconds  of  angle,  etc. 

Relations  of  identity  among  trigonometrical  functions  may 
be  proved  in  either  of  two  ways. 

FIRST  METHOD.  By  use  of  the  formulas  for  the  functions 
given  in  Arts.  28  and  29  (and  particularly  those  which 
reduce  the  function  to  sine  and  cosine)  an  expression  may 


TRIGONOMETRIC   FUNCTIONS  37 

be  proved  identical  with  another,  by  reducing  one  of  the 
given  expressions  directly  to  the  form  of  the  other. 

Ex.  1.     Prove  cot2  A  cos2  A  =  cot2  A  -  cos2  A. 

cos2  A 
sinz  A 

(1  —  sin2  A)  cos2  A 

sin2  A 

_  cos2  A      sin2  A  cos2  A 
sin2  A  sin2  A 

=  cot2  A  —  cos2  A. 

Instead  of  proving  an  identity  by  reducing  one  member  of 
the  identity  to  the  form  of  the  other,  it  is  sometimes  more 
advantageous  to  reduce  both  expressions  to  a  common  third 
form,  and  hence  infer  their  identity  by  Ax.  1. 

us  we  may  start  with  cot2  A  cos2  A  =  cot2  A  —  cos2  A  and  tr^ns- 
it  as  follows : 

cos2  A        o    -A      cos2  A          2  A 

—- —    COS2   A  =     .  —  COS2  A, 

snr  A  snr  A 

cos4  A  _  cos2  A  —  cos2  A  sin2  A 
sin2  A  sin2  A 

cos4  A  _  cos2  A(1L  —  sin2  A) 
sin2  ^4  sin2  A 

cos4  ^4  _  cos4  A 
sin2  ^4      sin2  A 

Since  the  last  is  plainly  an  identity,  we  infer  that 

cot2  A  cos2  A  =  cot2  A  —  cos2  A 
is  also  an  identity. 

SECOND  METHOD.  By  use  of  the  values  of  tiie  functions 
obtained  by  applying  the  definitions  of  the  functions  to  the 
right  triangle  (Art.  22,  Fig.  6). 

Ex.  2.     Prove  smA9  -  =  cot  A. 

cos  A  tan'5  A 


or 


38  TRIGONOMETRY 

Substitute  -  for  sin  A ;  -  for  cos  A ;  -  for  tan  A :  -  for  cot  A.     Then 
c  c  b  a 

a 
Bin  A  _L_6  =  Cot  A. 


cos  A  tan2  J.     ?>  a2     a 

EXERCISE  II 

Prove  each  of  the  following  identities  : 

(In  the  solution  of  identities,  the  first  of  the  two  methods  given  above  is  to  be 
preferred,  since  its  use  helps  fix  in  mind  the  fundamental  equations  and  formulas 
given  in  Arts.  28  and  29.) 

1.  cos  A  tan  A  =  sin  A.  5.    sin  A  =  cos  A  tan  A. 

2.  sin  A  sec  A  =  tan  A.  e. 


sin  A         1  —  cos  A 

3.   cos  A  esc  A  =  cot  A. 

1  +  sin  A         cos  A 


4.   cos  A  =  sin  A  cot  A.  cos  A         1  —  sin  A 

,  8.  sin2  A  —  cos2  A  =  2  sin2  A  —  \. 

9.  (1  -  sin2  A)  tan2 .4  =  sin2  A. 

10.  (tan  A  +  cot  ^4)  -sin  A  cos  A  =  l, 

11.  (1  —  sin2  ^4)  esc2  yl  =  cot2  A. 

12.  (sin  yl  +  cos^4)2  =  1  +  2  sin  ^4  cos  A. 

13.  (sin  J.  +  cos  ^4)2  +  (sin  J.  —  cos  yl)2  =  2. 

14.  (esc2 ,4  —  1)  sin2  A  =  cos2  A 

15. 

cos  A     sin 


1  +  cot2  A 

17.  tan  ^4  +  cot  A  —  sec  A  esc  ^4. 

18.  tan^  +  cot^=sec^  +  C8c2A 

sec^L  x  esc  A 

19.  sin4  A  —  cos4  ^4  =  sin2-  A  —  cos2  -4. 
o       sin  ^4  cos  -4 

1  —  cot  A     1  —  tan  A 


Ml  +  cos  A 


22. 


TRIGONOMETRIC   FUNCTIONS  39 

1  +  tan  A  __  1  —  tan  A 
l-|-.cot.l  cot  ^4  —  1 
1 


23.    cot  A  4-  tan  A  =  — 


sin  A  cos  ^4. 

24.  tan2  A  —  sin2  .1  =  tan2  A  sin2  .4 . 

25.  esc4 .1-2  esc2  A  ==  cot4  .1-1. 

26.  sec4  A  (I  —  sin4 .1)  =  2  tan2 .4+ 

27.  _J^A_         cosA 
tan  A  4-  cot  J. 

28.  ~cot2^  =  sin2  ^l  -  cos2.!. 
1  4-  cot-  A 

cot  -4  —  cos  A       cot  .1  cos  .1 


cot  A  cos  .1       cot  A  4-  cos  J 
30.   1- cot4 .1  =  2  csc2.! -csc4.!. 


31.  Vl  —  sin2  A  tan  A  =  sin  A. 

32.  sin6  A  4-  cos6 .1  =  1  —  3  sin2  A  cos2  A. 

.  cos3  A  —  sin3  A  =  (cos  A  —  sin  A)(\. 4- sin  .4  cos  A). 

[  Reduce  tan6  x  sec4  x  to  the  form  (tan8  x  4-  tan6  x)  sec2x. 

Transform : 

35.  tan8  x  into  (tan6  x  —  tan4  x  4-  tan2  x  —  1)  sec2  a?  4- 1 . 

36.  sec10 y  into  sec2?/  (1  4-  4  tan2?/  4-  6  tan4  y  4-  4  tan6  ?/  4-  tan8 y). 

GOSX 

37.  VI  4-  sin  x  into 


Vl  —  sin  x 

38.    — ; —  into  sec2  x  —  sec  x  tan  x. 

1  +  Sill  X 

1  4-  sin  x  •    ,          o 

39.  —         -  into  sec*  x  4-  sec  x  tan  #. 

cos2  a; 

40.  See  if  you  can  make  up  or  discover  any  other',  trigonometrical 
identities  for  yourself. 

41.  How  many  of  the  above  examples  can  you  work  at  sight? 

TRIGONOMETRIC   FUNCTIONS   OF   PARTICULAR  ANGLES 

32.  Functions  of  45°.  The  trigonometric  functions  of 
30°,  45°,  and  60°  are  used  so  frequently  that  it  is  of  service 
to  determine  their  values  and  commit  these  values  to 


40 


TRIGONOMETRY 


memory.      It  is  helpful  to  notice  that  we  determine  these 
values  in  each  case  by  the  use  of  a  right  angle,  the  hypote- 
nuse of  which  is  taken  as  1. 

Let  ABC  (Fig.  11)  be  an  isos- 
celes right  triangle,  the  hypotenuse  of 
which,  AB,  is  1.  Then,  by  geome- 
try, each  leg  is  lV2_(for  Z  B  =  45°, 
.-.AC^=  BC;  but  AC*  +  BC*  =  I2, 
FIG.  11.  •'•  2BC  =  I2,  etc.). 


C 


By  the  definitions  of  the  trigonometric  functions, 
sin  45°  = 
cos  45°  = 


cot  45°  = 


1V2 
•V2 


- 

1V2 


V9         9 

sec  45°  =l-_  =  -^ 
2       V2 


V2 


33.   Functions  of  30°  and  60°.     Let  ABD  (Fig.  12)  be  an 
equilateral  triangle  in  which  the  length 
of  one  side  is  1.     Let  AC  be  ±.BD. 

Then,  by  geometry 


6.0° "" 


and  Z  BAG  =30°. 

Also  .4  (7  bisects  BD,  hence 

AC  = 


FIG.  12. 


Then  in  the  right  triangle  ABC9 
sin  30°  =  1. 
cos  30°  =  1V3. 


TRIGONOMETRIC  FUNCTIONS 


41 


tan  30°  =  ^        =  -  =  =  1V3. 
V3      3 


cot  30°  = 


sec  30°  = 


1V3      V3 
esc  30°  =  T  =  2. 


Let  the  pupil  write  out  in  like  manner  the  functions  of 
60°  (that  is,  of  Z  ABC  in  the  A  ABC). 

Of  the  results  obtained  in  Arts.  32  and  33  those  which  are 
most  used  may  be  conveniently  arranged  i/i  a  table  thus: 


30° 

45° 

60° 

sin 

i 

IV2 

iV3 

cos 

iVs 

iVs 

i 

tan 

^Vs 

1 

V3 

34.  Functions  of  0°.  Let  ABC  (Fig.  13)  be  a  right 
triangle  in  which  the  hypotenuse  AB  =  1  and  the  angle 
BAG  is  small  and  is  diminished  and 
made  to  approach  0°  as  a  limit.  Then 
if  AB  remains  fixed  in  length,  BC  £ 
approaches  zero  and  AC  approaches  1. 

At  the  limit, 


FIG.  13. 


sin  0°  =  -  =  0. 
cos  0°  =     =  1. 


tan  0°  ==  ^  =  0. 


sec  0°  =  j  =  1. 
esc  0°  =  TT  =  GO. 
versO°=  1-1  =  0, 


cot  0°  =  -=•  =  oo. 


covers  0°  =  1  -  0  =  1 


42 


TRIGONOMETRY 


35.    Functions  of  90°.      Let   ABC   (Fig.  14)  be  a   right 
triangle    in    which    BAG    is    nearly    a    right    angle    and 
approaches  90°  as  a  limit.     AB  remains  fixed  in 
length;    hence   BC  approaches  1  as  a  limit  and 
AC  approaches  0. 


-C 


Fio.  14. 


At  the  limit. 


sin  90°  =  ±  =  1. 
cos  90°  =  y  =  0. 
tan  90°  =  TT  =  oo  . 
cot  9a°  =  ~  =  0. 


sec  90°  =  -  =  oo  . 
esc  90°  =  1  =  / 
vers  90°  -  1  -  0  -  1. 
covers  90°  =  1  -  1  =  0. 


The  results  obtained  in  Arts.  34  and 
35  may  be  conveniently  arranged  in  a 
table  thus : 


36.    Representation   of  the  Trigonometric  Functions  of  an 
Acute  Angle  by  Lines.     If   a  quadrant    of   a    circle     OAB 
be   drawn  with  center  0  and  radius  OB 
equal  to  1,  the  sine  of  any  angle  AOP'  is 
M'P'  =  M'P' 
1 


=  M'T. 


OP' 

Similarly   the   sine  of 
and  sine  of  Z  AOP"  =  M"P". 

In  other  words  the  sine  of  any  angle 
AOP  in  a  quadrant  whose  radius  is  1  is 
represented  by  the  perpendicular  let  fall  from  P  upon  the 
radius  OA. 


M"      M'  M'A 
FIG.  15. 


TRIGONOMETRIC   FUNCTIONS 


43 


Hence  it  is  easy  to  see. that,  since  MP  is  the  sine  of 
Z  A  OP,  if  AOP  becomes  very  small  and  =  0,  MP  =  0, 
and  at  the  limit  sin  0°  =  0.  Also  if  /.AOP"  increases 
and  =  90°,  sin  /_  AOP"  or  M"P"  =  OB  or  1.  Hence  at 
the  limit  sin  90°  =  1. 


Similarly  cos  Z  AOPf  = 


OM'      OM' 


=  OM!    Hence  also 


OP'         1 

cos  Z  AOP  =  OM,  cos  Z  AOP"  =  OJf."  In  other  words 
the  cosine  of  any  angle  A  OP  in  a  quadrant  whose  radius  is 
1  Dented  by  the  part  of  OA  intercepted  between  0 

and  uno  foot  of  the  line  representing  the  sine. 

Hence  cos  0°  =  0 A  or  1,  and  as  Z  A  OP  changes  from  0° 
to  90°,  the  cosine  changes  from  1  to  0. 

Similarly,  (Fig.  16), 

AT     AT 


AOT  = 


cot^AOT= 


OA 
OT 
OA 


1 
OT 


BR 
'OB 

OR 
OB 


BE 


=  OT. 


=  BR. 


o 


FIG.  16. 


OR 


N 


The  various  lines  which  represent 
the  trigonometric  functions  of  an  acute 
angle  AOP  may  be  combined  in  a 
single  figure  (Fig.  17).  Let  the  pupil 
find  the  lines  on  the  figure  which 
represent  vers  Z  A  OP  and  covers 
ZJ.OP. 

37.  Tables  of  Trigonometric  Functions  of  Angles  from  0°  to 
90°  called  Natural  Functions.  By  methods  which  will  be 
explained  later  (see  Art.  116)  the  values  of  the  trigonometric 


O 


M  - 
FIG.  17. 


44  TRIGONOMETRY 

functions  for  angles  of  every  degree  and  minute  from  0°  to 
90°  may  be  calculated.  These  values  are  arranged  in  tables 
called  Tables  of  Natural  Trigonometric  Functions. 

EXERCISE   12 

By  the  use  of  squared  paper,  construct  the  following  angles,  making 
use  of  their  natural  functions  :  • 

1.   30°.     (Use  sin  30°  =  1 )  2.   45°.  3.    60°. 

4.  If  tan  61°  37'  =  1.85,  construct  the  angle  61°  37'  on  squared  paper. 

By  use  of  the  table  of  natural  tangents,  construct : 

5.  42°  30'.  6.   56°  37'.  7.   47.24°.  8.   72.37°. 

By  use  of  the  table  of  natural  sines,  construct : 
9.   61°  23'.  10.   47°  15'.  11.   52.35°.  12.   63.84°. 

Find  the  numerical  value  of : 

13.  2  sin  30°  +  cos  60°  +  sin  90°. 

14.  b  tan  30°  +  c  cot  60°  +  «  tan  0°. 

15.  4  tan  0°  +  4  sin2  45°  -f  2  cos  45°. 

16.  tan  30°  cos  90°  -  4  sin  60°  +  cos2  0°. 

^  tan  30°  cot  30°  -  2  sin  45°  tan  45°  -  6  cos  60°  cot  45°  +  sin  90°. 

18.  sec  60°  cos  60°  -  tan  30°  cot  60°  +  tan  60°  cot  30°  -  20  sin  30°. 

19.  Show  that  (sin  60°  —  sin  45°)  (cos  30°  -f-  cos  45°)  =  i. 

*  If  P  =  0°,  Q  =  30°,  ^  =  45°,  £  =  60°,  T=90°,  find  the  value  of  each 
of  the  following  expressions : 

20.  sin  Q  +  cos  R  —  1. 

21.  tan2P  +  tan2Q-f-tan2^. 

22.  cos  P  cos  Q  cos  R  +  sin  R  sin  S  sin  T. 

23.  sec  P  +  2  sin  Q  +  2  cos2  R  +  £  tan2  S  +  cosec  T. 

24.  Does  twice  the  tangent  of  45°  =  the  tan  of  90°  ?     Why  ? 

25.  Does  sin  30°  -f  sin  45°  =  sin  75°? 

26.  Does  cot  30°  +  cot  45°  =  cot  75°  ? 

27.  Draw  a  diagram  showing  the  trigonometric  functions  as  lines 
when  Z  A  OP  is  less  than  45°. 

28.  Also  when  /-AOP  is  greater  than  45°. 

29.  Also  when  £AOP  equals  45.° 


TRIGONOMETRIC   FUNCTIONS  45 

30.  Given  that  x  is  greater  than  45°  and  less  than  90°, 'show  on  a 
diagram  similar  to  Fig.  17  that  tan  x  ifc  greater  than  cot  x. 

31.  Given  that  x  is  less  than  45°,  show  that  sec  x  is  less  than 
esc  x. 

32.  Show  that  cos  x  is  always  less  than  cot  x. 

33.  Show  that  sin  x  <  tan  x  <  sec  x. 

34.  Show  that  cot  x  <  esc  x. 

35.  If  a  flagstaff  is  at  a  distance  of  150  ft.  and  the  angle  of  elevation 
(see  Art.  88)  of  the  top  of  the  flagstaff  is  30°,  find  the  height  of  the 
flagstaff. 

36.  Find  its  height  if  the  angle  of  elevation  of  the  top  (at  the  same 
distance)  is  45°.     Is  60°. 

37.  Make  up  two  examples  similar  to  Ex.  35. 

38.  The  Washington  Monument  is  555  ft.  high.     At  a  certain  place 
the  angle  of  elevation  of  its  top  is  30°.     Find  the  distance  of  the 
monument  from  this  place. 

39.  At  a  certain  spot  165  ft.  from  the  top  of  a  particular  part  of 

*-a  Falls  the  angle  of  depression  (see  Art.  88)  of  the  bottom  of 
Is  is  45°.     What  is  the  perpendicular  extent  of  the  falls  ? 

40.  How  many  of  the  examples  in  this  exercise  can  you  work  at 
sight? 

38.    Many  trigonometric    equations    involving   only  acute 
angles  may  now  be  solved. 

Ex.  1.    Find  the  value  of  x  which  satisfies  the  equation 
sin  x  =  ^. 

Since  sin  30°  =  -*-,  in  the  given  equation  x  =  30°,  Ans. 

Ex.  2.    Solve  sinx  =  cosx. 

Dividing  each  member  by  cos  x,  tan  x  =  1. 

.-.  a  =  45°,  Ans. 

Ex.  3.    Solve  tan  x—  1  =  2  sin  a;  —  2  cos  x.  .. 

Substituting  for  tan  x,  ^HL?  _  1  =  2  sin  x  -  2  cos  x. 

COSX 

Hence,  sin  x  —  cos  x  =  2  sin  x  cos  x  —  2  cos2  x.  — 

Factoring,   (sin  x  —  cos  a;)(l  —  2  cos  x)  =  0. 

Hence,        sin  x  —  cos  x  =  0.     .-.  tano;  =  l,     x  =  45°. 

Also    1  —  2  cos  a;  =  0.     .-.  cos  x  =  %,  x  =  60°. 

Hence,  x  =  45°,  60°,  Ans, 


46 


TRIGONOMETRY 


Ex.  4.    Given  sin  x  =  cos  4  x,  find  x. 

By  Art.  26  we  may  substitute  for  sin  x  its  equal,  cos  (90°  —  x}. 

Then  cos  (90°  —  x)  =  cos  4  x. 


^  =  90°. 
x  =  18°, 


EXERCISE  13 
Solve  each  of  the  following  equations  : 

3.  cot  x'=  3  tan  x. 

4.  cot2  x  =  1 

o 

5.  Vl  —  sin2  x  —  1  +  sin  x. 


2  sin  ?/  -h  esc  ?/  =  3. 

13.  2  sin  x  V3  +  4  cos  x  =  5. 

14.  sec  x  =  2  tan  x. 

15.  4  sin2  x  —  tan2  x  =  cot2  x. 

16.  cot  a? -K2  tan  a? = 


7.  tan  x  4-  cot  x  =  2. 

8.  sec  x  =  V2  tan  x. 

9.  cos2  x  —  sin2  x  =  sin  x.   - 

11.  3  cot2  x  4-  cot  x  =  4. 
Solve : 

23.  sin  x  =  cos  5  x. 

24.  tan  y  —  cot  8  ?/. 

25.  cos  \x  —  sin  x. 


17.  3  cos#4- tanx  =  1  4-3  sino?. 

18.  tan#  =  2cot#  —  1. 

19.  esc  ?/  =  2  cot  y. 

20.  2  sin  x  +  cos  x  =  2. 

21.  2  sec  x  —  cosx  =  1. 

22.  sin2  x  4-  sin  a;  =  |. 

26.  sec  (45°  4-  #)  =  csc  a;. 

27.  sin  ?/  =  cos  ny. 

28.  sin  3  x  =  cos  2  x. 


29.  If  a  church  steeple  is  at  a  distance  of  80  ft.,  and  the  steeple  is 
80  ft.  high,  find  the  angle  of  elevation  of  the  top  of  the  steeple. 

30.  If  the  height  of  the  steeple  is  80.5  ft.  and  the  distance  of  the 
base  is  100  ft.,  see  if  you  can  find  the  angle  of  elevation  of  the  top  of  the 
steeple  by  use  of  the  table  of  natural  tangents  (pp.  91-96  of  the  tables). 

31.  Make  up  an  example  similar  to  Ex.  29. 

32.  Make  up  an  example  similar  to  Ex.  30. 

33.  In  a  right  triangle  given  c  =  62,  a  —  31,  find  A. 

34.  Given  c  =  150,  a  =  75,  find  B. 

35.  Given  c  =  120,  b  =  60  V3,  find  A. 

36.  How  many  of  the  examples  in  this  exercise  can  you  work  at 


A 


TRIGONOMETRIC   FUNCTIONS  47 

39.  Tables  of  Logarithms  of  the  Trigonometric  Functions 
from  0°  to  90°.  In  performing  numerical  work  involving 
trigonometric  functions,  it  is  usually  more  expeditious  to 
proceed  by  the  use  of  logarithms.  Hence  the  logarithms  of 
the  natural  trigonometric  functions  have  been  obtained  once 
for  all  and  arranged  in  tables  called  Tables  of  Logarithmic 
Trigonometric  Functions.  The  use  of  these  tables  is  ex- 
plained in  the  Introduction  to  the  Tables  (Arts.  7-11). 

EXERCISE  14 

By  the  use  of  five-place  tables,  find : 

1.  log  sin  26°  18'.  9.  log  sin  4°  6'  55". 

2.  log  cos  12°  16'.  10.  log  cos  17°  17'  30". 

3.  log  tan  36°  18'.  11.  log  cot  37°  28' 50". 

4.  log  cot  76°  18'.  12.  log  sin  78°  59' 30". 

5.  log  tan  55°  16'.  13.  log  tan  86°  46'  5". 

6.  log  tan  15°  18'.  14.  log  tan  4°  44'  50". 

7.  log  cos  86°  52'.  15.  log  cos  45°  48 '48". 

8.  log  tan  36°.  16.  log  cot  60°  52'  6". 

17.  We  have  proved  (see  Art.  33)  that  sin  30°  =  .5.     Obtain  log  .5 
and  thus  show  that  the  value  of  log  sin  30°  as  given  in  the  table  is 
correct. 

18.  Similarly  verify  the  value  of  log  sin  45°,  and  of  log  tan  60°,  as 
given  in  the  table. 

19.  In  the  rt.  A  ABC,  a  =  b  tan  A.      (Why  ?)      If  A  =  18°  16'  and 
b  =  18.63,  find  a. 

20.  In  the  rt.  A  ABC,  b  =  c  cos  A.      (Why  ?)     Find  b  if  c  =  18.675 
and  A  =  36°  36'  36". 


By  the  use  of  four-place  *  tables,  find  : 

21.  log  sin  15.3°.  24.  log  tan  78.8°. 

22.  log  cos  47.5°.  25.  log  sin  27.35°. 

23.  log  cot  33.7°.  26.  log  cos  26.36°. 

*  When  the  term  "four-place  tables  "  is  used  in  connection  with  angles,  the 
four-place  logarithmic  tables  for  the  decimally  divided  degree  are  meant.  See 
Arts.  18-19  of  the  tables. 


48  TRIGONOMETRY 

27.   log  tan  63.78°.  29.    log  cos  40.16°. 

28-   log  cot  12.65°.  30.   log  cot  29.23°. 

31.  In  the  rt.  A  BA C,  b  =  a  cot  A.     (Why  ?)      If  .4  =  18.67°  and 
a  =  .2167  feet,  find  6. 

32.  In  the  rt.  A  ABC,  a  =  c  sin  A      (Why?)      If  c  =  17.65  and 
A  =  59.72°,  find  a.     Also  find  6,  if  b  =  c  cos  A 

EXERCISE  15 
Using  five-place  tables,  find  A,  given : 

1.  log  sin  A  =  9.59632  -  10.  7.  log  cos  A  =  9.53390  - 10. 

2.  log  tan  A  =  9.73777  - 10.  8.  log  tan  A  =  1.06575. 

3.  log  cos  A  =  9.90951  - 10.  9.  log  sin  A  =  9.95788  -  10. 

4.  log  cot  A  =  10.07029  - 10.  10.  log  cot  A  =  1.02921. 

5.  log  sin  A  =  9.96159  -  10.  11.  log  sin  A  =  8.84501  - 10. 

6.  log  tan  A  =  0.44540.  12.  log  cos  A  =  8.84501  -  10. 


By  use  of  four-place  tables,  find  A,  given : 

13.  log  sin  A  =  9.6495  - 10.  20.    log  cos  'A  =  9.8409  -  10. 

14.  log  cos  A  =  9.8063  - 10.  21.   log  tan  A  =  0.2575. 

15.  log  tan  A  =  9.7384  - 10.  22.   log  cot  A  =  2.0248. 

16.  log  cot  A  =  0.4755.  23.   log  tan  A  =  1.5718. 

17.  log  cot  A  =  9.8248  -  10.  24.   log  sin  A  =  9.9596  -  10. 

18.  log  tan  A  =  0.4422.  25.   log  cos  A  =  9.3129  -  10. 

19.  log  cos  A  =9. 6351  -10.  26.   log  cot  A  =  0.5881. 

EXERCISE    16 

By  use  of  five-place  tables  find : 

1.  log  sin  0°  56'  18".  5.   log  cot  1°  18'  36". 

2.  log  tan  1°  16'  37".  6.   log  cos  89°  7'  19". 

3.  log  cos  88°  13' 26".  7.   log  sin  1°  6' 12". 

4.  log  tan  88°  54'  50".  8.   log  cot  88°  16'  32". 
Find  the  angle  A  if : 

9.   log  tan  ^1=7.88154 -10.  13.    log  tan  A  =  3.05992. 

10.  log  cos  A  =  8.28910  - 10.  14.   log  cot  A  =  2.88206. 

11.  log  sin  ^4  =  8.09600  -10.  15.   log  sin  A  =  6.88800  - 10. 

12.  log  cot  A  =  7.90390  - 10.  16.   log  cos  A  =  7.63702  - 10. 


TRIGONOMETRIC   FUNCTIONS 


49 


For  "angle  whose  log  sin  is"  we  may  write  "Z  log  sin,"  or  "antilog 


sin,"  hence  tind : 

17.  Z  log  sin  9.82627 -10. 

18.  Z  log  tan  10.90261  - 10. 
19^   Z  log  cos  9. 06000 -10. 
23.   In  the  A  ABC,  a  =  c  sin 

18'  48." 


20.  Z  log  cot  8.09599 -10. 

21.  Z  log  cos  8.09599 -10. 

22.  Z  log  tan  =  2.77651. 

Find  a  if  c  =  18.6  and  A  =  26° 


Find  the  value  of  the  following : 

528.7  x  cos  83°  16'  24"  x  tan2  75°  18'  24" 


24 


25. 


672  cot2 18°  32'  54"  x  sin  69°  -  cos2 15°  16'  34" 

265  x  tan  65°  18'  x  cos2 14°  28f  12" 
19  cot2 11°  16'  24"  x  sin  75°  15'  45"  x  .7* 


By  use  of  four-place  tables,  find : 

26.  log  cos  88.76°. 

27.  log  sin  0.762°. 

28.  log  cot  89.267°. 

29.  log  tan  1.067°. 

Find  angle  A  if : 

34.  log  cot  A  =  8.1067  -  10. 

35.  log  tan  A  =  8.2574  - 10. 

36.  log  cos  A  =  8.1360  — 10. 

37.  log  sin  A  =  8.0440  -  10. 

38.  log  tan  A  =  2.1080. 

39.  log  cot  A  =  2.0532. 

40.  log  sin  A  =  7.9100  —  10. 

41.  log  cos  A  =  7.9932  -  10. 

49.  In  the  rt.  A  ABC,  a  =  c  sin  A. 
and  A  =  1.267°. 

50.  In  the  rt.  A  ABC,  b  =  acot  A. 


30.  log  tan  88.763°. 

31.  log  cot  0.765°. 

32.  log  sin  1.267°. 

33.  log  cos  89.467°. 


Find: 

42.  log  cot  88.676°. 

43.  log  tan  88.676°. 

44.  Z  log  cot  8.1078 -10. 

45.  Z  log  tan  8.0295 -10. 

46.  Z  log  cos  8.0959  - 10. 

47.  Z  log  sin  8.0371  -  10. 

48.  log  tan  88.68°. 

(Why  ?)     Find  a  if  c  =  126.27, 

(Why  ?)     Find  b  if  a  =  0.4267, 


and  A  =  2.166°. 

632.7  x  cos  78.16°  x  tan2  71.62° 


51.   Find  the  value  of 


52.    Find  the  value  of 


426.8  x  sin  13.25°  x  cot2 12.47°  x  .8 
326  x  tan  38.25  x  cos2  88.627 
43  x  cot  0.826°  x  sin2  2.467°  ' 


50  TRIGONOMETRY 

EXERCISE  17.    REVIEW 

1.  In  the  right  A  ABC,  given  tan  A  =  T\  and  a  =  16,  find  b,  c,  and 
the  other  functions  of  A. 

2.  If  cos'^L  =  -?-,  find  the  value  of 


17  cos  A  —  cot  A 

3.  Show  that  cos  60°  cos  30°  +  sin  60°  sin  30°  =  cos  30°. 

4.  Show  that   cot  45°  +  CQt  90°  =1. 

1  -  cot  45°  cot  90° 

(Work  Exs.  5-12  without  the  use  of  tables.) 

5.  Which  is  greater,  sin  49°  or  cos  49°  ? 

6.  If  sin  A  =  f  ,  is  A  greater  or  less  than  45°  ? 

7.  If  tan  A  =  2,  is  A  greater  or  less  than  60°  ? 

8.  Which  is  the  greater,  tan  37°  or  cot  37°  ? 


9.   If  A  =  60°,  show  that  sin  1 A  =    /1 


10.    If  A  =  60°,  show  that  cot%A  =  <J  - 


cos  A 


cos  A 

11.  Which  is  greater,  sin  45°  or  %  sin  90°  ?     sin  60°  or  2  sin  30°  ? 
tan  30°  or  Han  60°? 

12.  If  x  =  30°  and  y  =  60°,  show  that  sin  x  cos  y  +  cos  x  sin  y  = 
sin  (a;  -f  y}> 

13.  Prove  1  +  cot  A  =  sec  ^  +  csc  A 

1  —  cot  A      sec  ^4  —  csc  A 

14.  Prove  X  +  ta"2 

1  +  cot 

COS 
COS 


15.  Prove  *  +  cos  ^  =  (csc  J.  +  cot  A)2. 

I  —  cos  A 

16.  If  x  =  30°,  show  that  tan  2x  = 


1  —  tan2  x 

17.  If  a;  =  30°,  show  that  sin  3  x  =  3  sin  x  —  4  sin3#. 

18.  If  x  =  30°,  show  that  cos  3  x  =  4  cos3  x  —  3  cos  x. 

Solve  the  following  trigonometric  equations  :  — 

19.  tan  x  +  3  cot  x  —  4. 

20.  2  sec2z-tan2x  =  5. 

21.  3csc2a;-2cota;  =  4. 


TRIGONOMETRIC   FUNCTIONS  51 

If  P  =  0°,  Q  =  30°,  R  =  45°,  S=  60°,  T  =  90°,  find  the  value  of : 

22.  cos2  Q  +  cos2  S  +  cos2  T  -f  2  cos  Q  cos  $  cos  T7. 

23.  sec  (2(1  +  tan  72)  —  sin3  T(cos  R  +  sin  #  cos  Q). 

24.  1  +  tan22  8  +  3(cos  P  sin2  fl  -  sin  S). 
A  —  tan  l\i 

25.  If  25  sin  A  =  7,  find  cot  ^4  and  esc  A 

26.  If  p  cot  (9  =  Vr2— i>2,  find  sin  0. 

27.  If  i  denotes  the  angle  of  incidence  of  a  ray  of  light  falling  on  a 
piece  of  glass,  and  r  the  angle  of  refraction,  then  sin  i  =  f  sin  r.     Find 
r  when  i'  =  27°  17'. 

28.  If  at  a  distance  of  300  ft.  the  angle  of  elevation  of  the  top  of 
one  of  the  big  trees  of  California  is  45°,  how  tall  is  the  tree? 

29.  If  at  a  distance  of  300  ft.  the  angle  of  elevation  of  the  top  of  a 
tree  were  42°,  see  if  you  can  find  out  how  tall  the  tree  would  be.     (Why 
are  we  able  to  determine  this   height  by  trigonometry  and  not   by 
geometry  ?) 

30.  Who  first,  and  at  what  date,  defined  the  sine  of  an  angle  as  the 
ratio  between  two  lines  (see  p.  165)  ?     Give  the  different  substitutes 
for  this  idea  of  the  sine  that  had  been  used  before  this  time.     Why  is 
the  ratio  definition  of  the  sine  superior  to  each  of  these  ? 

31.  Explain  the  origin  and  literal  meaning  of  the  word  sine  (see 
p.  166). 

32.  Who  first  invented  each  of  the  other  trigonometric  ratios,  and 
at  what  time  (see  pp.  162,  164)  ? 

33.  Give  some  of  the  various  names  used  for  these  ratios,  with  the 
names  of  the  inventors  of  these  names. 

34.  What  nation  first  used  the  trigonometrical  identity 

sin2  A  +  cos2  A  =  l  (see  p.  172)  ?     tan  x  =  ^^  ? 

cosx 

35.  Give  an  account  of  the   computation  of   trigonometric   tables 
(see  pp.  168-170). 


CHAPTER   III 
RIGHT   TRIANGLES 

40.  Two    Cases   arise  in  the  trigonometrical   solution  of 
right  triangles. 

CASE  I.     Given  one  side  and  an  acute  angle. 
CASE  II.     Given  two  sides. 

In  each  of  these  cases  it  will  be  observed  that  three  parts  are  really 
given,  since  the  right  angle  is  known. 

CASE  I 

41.  The  solution  of  Case  I  is  effected  as  follows : 

Subtract  the  given  angle  from  90°.  This  ivill  give  the  un- 
known angle. 

The  unknown  sides  may  then  be  found  by  means  of  the 
following : 

1.  Either  leg  =  (sine  of  ^opposite)  x  hypotenuse. 

2.  Either  leg  =  (cosine  o/Z  adjacent)  x  hypotenuse. 

3.  Either  leg  =  (tangent  of/,  opposite]  x  other  leg. 

4.  Hypotenuse  =  (secant  of  either  acute  Z)  x  (leg  adjacent 
to  thatZ.}. 

Also  (either  leg)  =  (cot  of  Z adjacent)  x  (other  leg); 

hyp.  =  (esc  of  either  acute  Z)  x  (leg  opposite  that  Z). 

Proof 


By  def .,      sin  A  =  -• 

c 


A          b          C 
FIG.  18, 


Also 


Also 


cos  B  =  -. 

c 

tan  A  =  -- 
b 


sec  B  =  - 
a 

52 


,  a  =  c  sin  A. 
a  =  c  cos  B. 
a  =  b  tan  A. 
.  c  =  a  sec  jB. 


RIGHT   TRIANGLES 


53 


Similarly  it  may  be  proved  that : 

b  =  c  sin  jB,  b  =  c  cos  A,  b  =  a  tan  B,  and  c  =  b  sec  A. 

Ex.  1.     Given  A  =  55°  43'  29",  c  =  415.18,  find  the  remain- 
ing parts  of  the  right  triangle. 

We  first  draw  a  diagram  (Fig.  19)  of  the  triangle  to  be  solved,  and  on 
this  diagram  write  the  known  magnitudes  (415.18  for  c,  and  55°  43'  29" 
for  A).  We  also  indicate  the  parts  to  be  computed  (a,  b,  B)  by  annex- 
ing the  =  mark  to  each  of  these.*  During  the  numerical  computation, 
as  soon  as  the  result  for  any  part  is  ascertained,  this  result  should  be 
entered  on  the  diagram  after  the  proper  =  mark. 
Z  B  =  90°  -  55°  43'  29"  =  34°  16'  31 ". 

a  =  415.18  sin  55°  43'  29".         (Art.  41,  1) 
.-.  log  a  =  log  415.18  +  log  sin  55°  43'- 29". 

415.18  log       2.61824 
55°  43'  29"  log  sin  9.91716  - 10 
a  =  343.085      log        2.53540 
Also  b  =  415,18  cos  55°  43'  29".         (Art.  41,  2) 
.-.  log  b  =  log  415.18  -f  log  cos  55°  43'  29". 

415.18  log      2.61824 
55°  43'  29"  log  cos  9.75064  - 10 


=  233.821       log       2.36888- 
(As  a  check  use  a  =  b  tan  A.) 


0 


FIG.  19. 


Ex.  2. 


A  b 

FIG 


Given  a=  .0723,  ^  =  31°  47'  7",  find  the  remain- 
ing parts  of  the  right  triangle. 

Z  A  =  90°  -  31°  47'  7"  =  58°  12'  53". 
b=    .0723  tan  31°  47' 7" 

.0723      log         8.85914  - 10 
31°  47' 7"  log  tan  9.79216  - 10 
b  =  .448022     log        8.65130  - 10 
c=. 0723  sec  31°  47' 7" 

.0723 
cos  31°  47"  7' 

5723  log  8.85914  - 10 
31°  47'  7"  log  cos  9.92943  -  10  colog  cos  0.07057 

c=        .0850567      "Tog        8.92971  - 10 
(As  a  check  use  b=  c  cos  A.) 


=         C 

.  20. 


54  TRIGONOMETRY 

Ex.  3.     By  use  of  four-place  tables  solve  the  right  triangle 
in  which  &  =  21.635,  .A  =  47.23°. 

Z  B  =  90°  -  47.23°  =  42.77°. 
Also          a  =  21.635  tan  47.23°.  (Art.  41,  3) 

B          .'.  log  a  =  log  21.635  +  log  tan  47.23°. 
21.635  log        1.3352 
47.23°  log  tan  0.0339 
a  =23.394  log        1.3691 
By  Art.  41,  4,  c  =  21.635  sec  47.23°  = 


cos  47.23° 

.-.  log  c  =  log  21.635  H-colog  cos  47.23° 
'21j635        — 'C  21.635      log      1.3352 

FIG.  21.  47.23°  colog  cos  0.1681 

c  =  31.864      log      1.5033 
(As  a  check  use  a  =  c  cos  J5.) 

42.  First  Estimates.  Graphical  Solutions.  In  the  solutions 
of  triangles  fully  one  half  the  mistakes  commonly  made,  and 
those  the  most  important  ones,  are  eliminated  by  making  a 
rough  mental  forecast  of  the  results  before  proceeding  with 
the  exact  numerical  work. 

Thus  in  solving  Ex.  1  of  Art.  41,  the  pupil  should  first  of  all  observe 
that,  the  hypotenuse  being  415.18,  each  of  the  legs  will  be  less  than 
415.18 ;  and  also  that,  since  angle  B  is  less  than  angle  A,  side  b  must 
be  less  than  side  a.  If  then  as  a  result  of  his  exact  numerical  calcula- 
tion, the  pupil  finds  a  leg  greater  than  415.18,  or  a  less  than  6,  he  knows 
at  once  that  a  mistake  has  been  made. 

Similarly  it  is  useful,  by  means  of  the  rule  and  protractor, 
to  make  a  drawing  according  to  scale  of  the  triangle  to  be 
solved,  and  from  the  figure  to  determine  as  accurately  as 
possible  the  dimensions  of  the  unknown  parts  by  measuring 
them  according  to  scale.  Such  results  should  be  accurate 
enough  to  aid  in  eliminating  any  large  errors  in  the  numeri- 
cal work.  (Indeed,  if  the  work  be  neatly  done,  the  results 
obtained  from  the  diagram  will  be  accurate  enough  for  many 
practical  purposes.) 


RIGHT   TRIANGLES 


55 


43.    Exact  checks  of  the  numerical  accuracy  of  the  work 

of  solving  right  triangles  are  obtained  by  calculating  some 
side  or  angle  of  the  triangle  by  a  formula  different  from 
those  already  used  in  the  computation,  and  observing  whether 
the  results  thus  obtained  accord  with  those  obtained  in  the 
first  solution. 

Thus,  to  check  the  accuracy  of  the  solution  given  for  Ex.  1,  Art.  41, 

determine  whether  tan  A  =  --,  that  is,  compute  the  value  of  the  frac- 

b 

tion  343>Q85  and  also  obtain  from  the  table  the  value  of  tan  55°  43'  29" 
233.821 

and  observe  whether  these  two  values  accord. 

EXERCISE  18 

State  at-  sight  the  formula  value  of  x  (or  of  x  and  y)  in  each  of  the 
following  triangles : 

Thus  in  Ex.  1,  (1),  x  =  208  sin  40°. 
1. 
(3) 


(1) 


(2) 


3.  Make  up  an  example  similar  to  Ex.  2. 

By  use  of  five-place  tables  solve  each  of  the  following  triangles,  given  : 
(In  working  each  example  outline  all  the  work  carefully  before  looking  up 
any  logs  — see  Ex.  1,  p.  18.) 

4.  .4  =  28°,   6  =  12.  6.   .4  =  46°  18',   6  =  48.527. 

5.  -4  =  78°,'  c  =  26.736.  7.   .4  =  28°  17',    c  =  24.16- 


56  TRIGONOMETRY 

8.  B  =  54°  43'   c  =  1123.  10.    A  =  38°  16'  24",    c  =  3.6289. 

9.  B  =  37°  19',   6  =  293.8.  11.    B  =  72°  16'  42",   a  =  22.684. 

12.  Given    c  =  . 52684,   B  =  63°  18'  48";    find  a. 

13.  Given  A  =  37°  25'  20",   c  =  .356 ;    find  b. 

Find  the  remaining  parts  in  each  of  the  following  right  triangles, 
given : 

14.  ^  =  63°  28'  40",   a  =  256.43. 

15.  c  =  13.867,   A  =  87°  16'  30". 

16.  A  =  51°  9'  6",   c  =  .19678. 

17.  a  =  126.78,   A  =  26°  18'  36". 

18.  Given  ^4  =  5°  16'  32",   b  =  .96156;    find  c. 

19.  Given  A  =  37°  14'  15",   b  =  217 ;    find  a. 

20.  If  the  top  of  the  Statue  of  Liberty  in  New  York  harbor  is  301 
ft.  above  the  water  surface,  and  a  boat  in  the  harbor  finds  the  angle  of 
elevation  of  the  top  of  the  statue  to  be  12°,  how  far  is  the  boat  from 
the  statue  ? 

21.  If  a  certain  point  on  the  brink  of  the  Grand  Canon  of  the  Colo- 
rado is  known  to  be  a  horizontal  distance  of  3  miles  from  the  Colorado 
River  and  the  angle  of  depression  of  the  river  is  17°,  how  deep  is  the 
canon  at  that  place  and  how  far  from  the  observer  is  the  river  in  a 
straight  line? 

22.  Which  of  the  examples  in  Exercise  22  are  you  able  to  solve  by 
Case  I  ?     Solve  one  of  these. 

23.  Make  up  a  similar  practical  problem  for  yourself  and  solve  it, 
as  for  instance  one  concerning   the  Bunker  Hill  monument  (221  ft. 
high). 

Solve  the  following  right  triangles,  by  use  of  four-place  tables,  hav- 
ing given  : 

24.  .4  =  32.6°,    6  =  18.  28.  .4=*37.67°,  c  =  126.7. 

25.  .4  =  56°,   c  =  2.678.  29.  £=.76.25°,  a  =  .926. 

26.  5  =  38.2°,   c  =  .7685.  30.  .4  =  21.32°,  a  =  16.256. 

27.  5  =  82.5°,   a  =  12.56.  31.  5=66.27°,  b  =  .0087. 

32.  Given    c  =  .6243,   5  =  51.25°;    find  a. 

33.  Given  A  =  77.26°,   c  =  .5163;    find  b. 
^34.    Given  5  =  39.29°,   6  =  41.67;    find  a. 


RIGHT   TRIANGLES  57 

Find  the  remaining  parts  in  each  of  the  following  right  triangles, 
given : 

35.   c  =  13.13,  A  =  88.17°.  36.   5  =  42.16°,   a  =  .5252. 

37.  Given  A  =  5.26°,  6  =  128.6;    find  c. 

38.  Given  B  =  87.267°,   c  =  22.67 ;  find  a. 

39.  Given  A  =  4.276°,    a  =  26.32 ;  find  6. 

40.  Work  Exs.  20-23  by  four-place  tables. 


Solve  without  the  use  of  tables,  having  given : 

41.  .4  =  30°,   b  =  7.  45.   .4  =  60°,   a  =  2000. 

42.  .4  =  45°,   c  =  12.  46.    J5  =  30°,   c  =  1200. 

43.  5  =  60°,   6  =  25.  47.   ^4  =  45°,   6  =  200. 

44.  5  =  30°,   a  =  1000.  48.   .4  =  30°,    c  =  20d. 

49.  Solve  Exs.  6  and  7  of  this  exercise  without  the  use  of  logarithms 
(i.e.  by  the  use  of  the  Tables  of  Natural  Sines,  etc.,  pp.  91-96). 

50.  How  many  of  Exs.  41-48  can  you  solve  at  sight  without  draw- 
ing a  figure  ? 

51.  On  the  figure  if  AADB  and  DOB  are 
right  A,  find  BD,  BC,  and  DC  at  sight. 

52.  On  Fig.  52,  p.  93,  if  OP  =  1,  what  is 
the  value  of  OQ  ?   of  PQ?  of  QN?  of  ON? 

CASE  II 

TWO  SIDES   GIVEN 
44.    The  Solution  of  Case  II  is  effected  as  follows: 

Find  one  of  the  angles  of  the  given  triangle  by  using  that  one 
of  the  following  trigonometric  ratios  which  contains  the  two  given 
sides : 

1.  sine  of  either  acute  £  =  -7 • 

hyp. 

2.  cosine  of  either  acute  ^  =  -r • 

3.  tangent  of  either  acute  ^-  =  -. — -TT-. 

•i  aa]. 


58 


TRIGONOMETRY 


Find  the  remaining  parts  of  the  triangle  by  Case  I  (but 
if  the  hypotenuse  and  a  leg  are  given,  the  other  leg  may 
be  found  by  one  of  the  formulas,  a  =  V(c  +  b)(c  —  b), 
b  =  V(c.+  a)(c-  a)). 

Ex.  1.  Given  a  =  317,  c  =  438,  find  the  remaining  parts 
of  the  right  triangle  ABC. 

sin  A  =  — —  (Art.  44,  1) 

B  Hence  log  sin  A  =  log  317  +  colog  438 

317  log  2.50106 

438  log  2.64147  colog  7.35853  -  10 
A  =  46°  21'  55"  log  sin  9.85959  -  10 
B  =  90°  -  46°  21'  55"  =  43°  38'  5". 
b  =  438  cos  46°  21'  55".      (Art.  41,  2) 

438  log  2.64147 
46°  21'  55"  log  cos  9.83888  - 10 


FIG.  22. 


b  =  302.24  log  2.48035 

(As  a  check  use  tan  A  =  —  • } 
b  J 


Ex.  2.     By  use  of  four-place  tables,  solve  the  right  triangle 
in  which  a  =  3.104,  £  =  2.965. 


A 

3.104 
2.965 

~2.965 
c-  3'104 

c/ 

(Art.  41)        ^/ 

cos  B 

log        0.4920 
colog     9.5279  -  10 

A 

3.104 
43.69° 

2.965                 G 
FIG.  23. 

log            0.4920 
colog  cos  0.1408 

=  46.31°  log  tan  0.0199 
=  90°  -  46.31°  =  43.69°. 

c=   4.293 

log 

0.6328 

45.  Sources  of  Power  in  Trigonometrical  Solution  of  Tri- 
angles. There  is  danger  that  the  pupil  form  mechanical 
habits  of  solving  triangles  without  realizing  the  nature  or 


RIGHT   TRIANGLES  59 

meaning  of  what  he  is  doing.  He  should  constantly  realize 
that  he  is  able  to  do  what  he  is  doing  because  some  one  be- 
fore him  has  computed  the  legs  of  every  possible  right  tri- 
angle whose  hypotenuse  is  1,  and  the  other  parts  when  each 
leg  is  1,  and  arranged  the  results  in  tables  (natural  sines, 
etc.,)  and  that  he  uses  these  results  (and  therefore  uses  the 
work  done  in  computing  them)  by  the  geometrical  principle . 
of  similar  triangles.  Also  that  some  one  else  has  made  the 
pupil's  work  easier  by  looking  up  the  logarithms  of  all  the 
numbers  in  the  natural  tables  and  arranging  them  in  other 
tables,  and  that  the  pupil  is  using  this  work  also. 

46.  Special  Case.  Given  the  hypotenuse  and  a  leg  nearly 
equal,  the  angle  between  them  will  be  very  small.  If 
this  angle  be  found  directly  from  the  parts  given,  it  will  be 
found  in  terms  of  the  cosine.  Since  the  cosine  of  a  small 
angle  changes  slowly  as  the  angle  varies,  such  a  solution  will 
not  be  accurate  in  the  last  figures.  A  more  accurate  solution 
is  obtained  by  first  calculating  the  third  side  by  the  use  of 
the  formula  a  =  ^/(c  +  b)(c  —  b)  and  finding  the  angle  men- 
tioned in  terms  of  the  sine. 

Ex.     Given  c=  412,  b  =  410,  solve  the  triangle. 


By  the  formula,  a  =  V(412  +  410) (412  -  410) 

=  V822  x  2. 
.-.  log  a  = -I-  (log  822  + log  2). 

822  log  2.91487 
2  log  0.30103 

2)3.21590 

A1  4      40.546 

a  =  40.546  log  1.60795  Also  sm  A  :  : 

40.546  log  1.60795 
412  colog  7.38510  -  10 


A  =  5°  38'  52"  log  sin  8.99305  -  10 
B  =  90°  -  5°  38'  52"  =  84°  21'  8". 


60  TRIGONOMETRY 

EXERCISE  19 

Using  five-place  tables,  solve  in  full  the  following  right  triangles, 
given : 

(In  working  each  example  outline  all  the  work  carefully  before  looking  up 
any  logs  —  see  Ex.  1,  p.  18.) 

1.  c=18.4,   a  =  10.7.  5.  c  =  . 89672,   a  =.68425. 

2.  c  =  37.266,   a  =  20.46.  6.  6  =  14.222,   c  =  21.678. 

3.  a  =  26.725,   c  =  39.626.  7.  a  =  .0628,   b  =  .0487. 

4.  a  =  5,    6  =  6.  8.  a  =  .1777,    c  =  . 25643. 

9.  Given  a  =  4  yd.,   6  =  9  ft.,  find  A. 

10.  Given  a  =  8.701  yd.,  b  =  21.645  yd.,  find  Z  A. 

11.  Given  b  =  .26725,   c  =  .39626,  find  Z  B. 

12.  Solve  in  full  if  a  =  6,    6  =  6. 

13.  Find  A  if  a  =  .02678,   6  =  .05537. 

14.  Solve  in  full  if  c  =  117.32,   a  =  112.67. 


SUGGESTION.     First  use  6  =  Vc2  —  a2  =  V(c  +  a)  (c  —  a). 

15.  Solve  in  full  if  b  =  358,   c  =  362. 

16.  Solve  in  full  if  a  =  26.63,   c  =  27.99. 

17.  If  the  Mt.  Washington  railway  at  a  certain  place  rises  3596  ft.  for 
3  mi.  of  the  length  of  the  track,  what  angle  on  the  average  does  the 
track  make  with  the  horizon  ? 

18.  The  carpenter's  rule  for  constructing  J  of  a  right  angle  is  to  con- 
struct a  right  triangle  whose  legs  are  5  and  12  inches  and  take  the 
greater  acute  angle  in  the  triangle.    How  far  is  this  from  being  correct  ? 

19.  Which  of  the  examples  in  Exercise  22  are  you  able  to  solve  by 
the  methods  of  Case  II  ?     Solve  two  of  these. 

20.  Make  up  a  similar  practical  problem  for  yourself  and  solve  it. 


Solve  by  use  of  four-place  tables,  having  given : 

21.  c  =  23.7,   a  =  15.7.  25.    6  =  6.7,    c  =  9.7. 

22.  c  =  .562,   6  =  .3962.  26.    6  =  .12675,  a  =  .14296. 

23.  a  =  33.29,   6  =  27.28.  27.    c  =  132.96,   6  =  100.82. 

24.  a  =  5,   6  =  8.  28.    a  =  .07282,   c  =  .11111. 

29.   a  =  2367,   6  =  1827.6. 


RIGHT   TRIANGLES 


61 


30.  Given  a  =  11,    c  =  16,  find  A. 

31.  Given  a  =  27.82,   b  =  33.67,  find  B. 

32.  Given  c  =  156.7,   b  =  148.2,  solve  in  full. 
First  use  a  =  Vc2  -  62  =  Vc  +  6)(c-6). 

33.  Given  c  =  862,   a  =  854,  solve  in  full. 

34.  Given  a  =  98.6,    b  =  63.4,  find  A. 

35.  Given  c  =  .4367,  b  =  .1967,  find  5. 

36.  Work  Exs.  17-20  by  the  four-place  tables. 


Without  the  use  of  tables  solve  in  full  each  of  the  following  right 
triangles,  given : 

37.  a  =  13,   6  =  13.  41.  c  =  6,   a=3V3. 

38.  c  =  18,   a  =  9.  42.  c  =  V2,    6  =  1. 

39.  c  =  200,   6  =  100.  43.  c  =  100,   a  =  50V3. 

40.  a=V3,   6  =  1.  44.  a  +  c=18,  6  =  6V3. 

45.  Solve  Exs.  3  and  4  of  this  Exercise  without  the  use  of  logarithms. 

46.  How  many  of  Exs.  37-43  are  you  able  to  solve  at  sight  without 
drawing  a  figure  ? 

47.  Isosceles    Triangles.     If  certain    parts  of  an  isosceles 
triangle  be  given,  the  unknown  parts  may  often  be  deter- 
mined by  dividing  the  isosceles  triangle  into  two  equal  right 
triangles  by  means  of  a  perpendicular  drawn  from  the  vertex  to 
the  base,  and  by  solving  one  of  the  right  triangles  thus  formed. 

Ex.  1.    If  the  vertex  angle  of  an  isosceles  triangle  is  42°  30' 
and  a  leg  is  47.6,  find  the  base. 

Draw  the  altitude  OD.    Then  /.A  OD=2l°  15'. 
Hence,  in  the  right  A  AOD,  we  have  a  side 
and   an   acute    angle    given,   to    find   the   base 
AD  (Case  I).  Hence 

AD  =  47.6  sin  21°  15'. 

47.6  log  1.67761 
21°  15' log  sin  9.55923 -10 
AD  =  17.252  log  1.23684 

AB  =  2  AD  =  34.501  FIG.  25. 


B 


62 


TRIGONOMETRY 


Ex.  2.    By  use  of   four-place   tables,  solve   the   isosceles 
A  triangle  whose  base  is  12.25  and  vertex  angle 

28.22°. 

Draw  the  altitude  AD. 

Then  Z  BAD  =  1(28.22°)  =  14.11°. 

Z  B  =  90°  - 14.11°  =  75.89°. 


AB  =  6.125  sec  75.89°  = 


6'125 


6.125 
FIG.  26. 


cos  75.89 
6.125  log  0.7872 
75.89°  colog  cos  0.6130 
AB  =  25.129  log  1.4002 


48.  A  regular  polygon  may  be  divided  into  equal  right  tri- 
angles by  lines  drawn  from  the  center  to 
the  vertices  and  by  the  apothems  to  the 
sides.  Hence  if  certain  parts  of  a  reg- 
ular polygon  are  given,  the  remaining 
parts  may  often  be  determined  by  divid- 
ing the  polygon  into  right  triangles  and 
solving  one  of  these  triangles. 

It  is  to  be  observed  that  one  of  the 
right  triangles,  as  A  CD  of  Fig.  27,  has 
the  radius  of  the  circle  circumscribed  about  the  polygon  for 
its  hypotenuse  AC,  and  the  radius  of  the  inscribed  circle, 

360° 

CD.  for  a  leg.     Hence,  Z  AC  A '=  -   — ,  where  n  denotes  the 

n 

number  of  sides  of  the  polygon,  and  Z  A  CD  of  the  right 

180° 

triangle  = . 

n 

EXERCISE  20 

Using  five-place  tables,  solve  each  of  the  following  isosceles  triangles, 
given : 

1.  Base  =  120,  base  Z  =  60°. 

2.  Leg  =  216,  vertex  Z=  110°. 

3.  Base  Z  =  56°  18',  leg  =  8.7265. 

4.  Base  Z.  =  38°  17'  50",  altitude  =  31.42. 


RIGHT   TRIANGLES  63 

5.  Base  Z  =  55°  18'  24",  altitude  =  762.89. 

6.  Base  =  8.2364,  altitude  =  7-8. 

7.  Vertex  Z  =  113°  17',  base  =  .12692. 

8.  Altitude  =  4835,  base  =9248. 

9.  One  side  of  a  regular  pentagon  is  12.     Find  the  apothem,  radius. 
perimeter,  and  area  of  the  pentagon. 

10.  One  side  of  a  regular  decagon  is  1.     Find  the  apothem,  radius, 
perimeter,  and  area  of  the  decagon. 

11.  The  radius  of  a  circle  is  16  feet.     Find  the  side,  apothem,  and 
area  of  a  regular  inscribed  dodecagon. 

12.  Find  the  same  magnitudes  for  a  regular  dodecagon  which  is 
circumscribed  about  a  circle  whose  radius  is  17.. 

13.  The  diagonal  of  a  regular  pentagon  is  14  ;  find  the  side,  apothem, 
perimeter,  and  area  of  the  pentagon. 

14.  The  apothem  of  a  regular  heptagon  is  0.69786  ;  find  the  perimeter 
and  area  of  the  heptagon. 

If  m  denotes  the  base,  li  the  altitude,  I  the  leg,  C  the  vertex  angle, 
and  D  the  base  angle  of  an  isosceles  triangle,  find  : 

15.  7i,  m,  and  (7,  in  terms  of  D  and  I. 

16.  D,  I,  and  (7,  in  terms  of  m  and  h. 

17.  D,  C,  and  m,  in  terms  of  h  and  I. 

18.  Cy,  h,  and  /,  in  terms  of  D  and  m. 

19.  D,  h,  and  /,  in  terms  of  C  and  m. 

20.  Solve  the  isosceles  triangle  in  which  a  leg  =  2.  62731  and  the 
altitude  =  1.76683. 

21.  If  a  chord  22.67  ft.  in  length  subtends  an  arc  127°  23',  what  is 
the  radius  of  the  circle  ? 

22.  If  the  radius  of  a  circle  is  105.27  ft.,  what  is  the  length  of  a 
chord  which  subtends  an  arc  of  54°  13'  ? 

23.  The  side  of  a  regular  polygon  of  fourteen  sides  inscribed  in  a 
circle  is  21.6  ft.  ;  find  the  side  of  a  regular  twenty-sided  polygon  in- 
scribed in  the  same  circle. 

24.  The  radius  of  a  circle  is  R,  show  that  each  side  of  a  regular 


inscribed  polygon  of  n  sides  is  2  R  sin  I-   -J,  and  that  each  side  of  a 

/180°\ 
regular  circumscribed  polygon  is  2  E  tan  (  -   -  )• 


64  TRIGONOMETRY 

\/         25.   Each  side  of  a  regular  polygon  of  n  sides  is  m;  show  that  the 
radius  of  the  circumscribed  circle  is  equal  to  —  esc  (  -    — ) ,  and  the  radius 

of  the  inscribed  circle  is  equal  to  —  cot  f ]• 

2         \  n   J 

26.  If  the  chord  of  an  arc  of  36°  is  24,  find  the  chord  of  an  arc  of 
12°  in  the  same  circle. 

27.  If  the  chord  of  an  arc  of  48°  is  36,  find  the  chord  of  an  arc  of 
66°  in  the  same  circle. 


Using  four-place  tables,  solve  the  isosceles  triangle  in  which  : 

28.  Leg  =  36.72,  base  Z  =  32.6. 

29.  Base  =  1600,  base  Z  =  67.4°. 

30.  Vertex  Z  =  117.72°,  altitude  =  17.83. 

31.  Base  =  .7368,  altitude  =  .4864. 

32.  Altitude  =  112.67,  leg  =  128.7. 

33.  Leg  =  67.87,  base  Z  =  32.73°. 

34.  Altitude  =  .11683,  base  Z  =  76.18°. 

35.  Base  =  31.26,  altitude  =  21.73. 

36.  Vertex  Z  =  151.7°,  leg  =  .4363. 

37.  One  side  of  a  regular  octagon  is  14.     Find  the  apothem  and 
area  of  the  octagon. 

38.  The  apothem  of  a  regular  pentagon  is  19.7.     Find  the  perimeter 
of  the  pentagon. 

39.  A  regular  decagon  is  inscribed  in  a  circle  whose  radius  is  1.76. 
Find  the  side  and  apothem  of  the  decagon. 

40.  Find  the  magnitude  of  the  various  parts  of  a  regular  heptagon 
circumscribed  about  a  circle  whose  radius  is  21. 

41.  The  diagonal  connecting  two  alternate   vertices   of  a   regular 
dodecagon  is  18.     Find  the  side,  apothem,  and  area  of  the  dodecagon. 

42.  If  a   chord   of  37.82  ft.   subtends   an   arc  of   118.3°,  find   the 
radius  of  the  circle. 

43.  If  the  radius  of  a  circle  is  100,  what  is  the  length  of  a  chord 
which  subtends  an  arc  of  67.7°  ? 


RIGHT   TRIANGLES  65 

Without  the  use  of  the  tables,  solve  the  following  : 

44.  The  base  of  an  isosceles  triangle  is  50,  and  the  vertex  angle  is 
120°.     Find  the  base  angle  and  altitude. 

45.  The  leg  of  an  isosceles  triangle  is  100,  and  the  altitude  is  50. 
Find  the  base  angle  and  base. 

46.  The  altitude  of  an  isosceles  triangle  is  10,  and  the  base  angle  is 
60°.     Find  a  leg  and  the  base. 

47.  The  leg  of  an  isosceles  triangle  is  6V2,  and  the  base  is  12.   Find 
the  base  angle,  vertex  angle,  and  altitude. 

48.  The  radius  of  a  circle  is  2.     Find  the  number  of  degrees  in  an 
arc  which  subtends  a  chord  whose  length  is  2V3. 

49.  The  diagonal  of  a  square  is  10.     Find  the  side  of  the  square. 

50.  How  many  of  Exs.  44-49  can  you  work  at  sight  ? 

AREAS 

49.  General  Method  of  computing  Area  of  a  Right  Triangle. 
If  1}  denote  the  base,  a  the  altitude,  and  K  the  area  of  a 
right  triangle,  by  geometry  K  =  ^db. 

.'.  log  K  =  log  a  +  log  b  +  colog  2. 

Ex.  1.  Given  J.  =  37°19',  6=308,  find  the  area  of  the 
right  triangle. 

To  find  log  a  and  then  the  area  we  proceed 

as  follows :  B 

a  =  308  tan  37°  19'.  (Art.  41) 

308  log  2.48855 
37°  19'  log  tan  9.88210  - 10 

a  log  2.37065  •/       K= 

308  log  2.48855 
2  colog  9.69897  -  10 
.fiT  =  36155  log  4.55817  FIG.  28. 

Ex.  2.  Find  the  area  of  a  right  triangle  in  which  the 
hypotenuse  is  417  and  the  base  356. 


a  =  Vtf^b2  =  V(417)2 -  (356)2 
=  V(417  +  356)(417-356)  =  V773  x  61. 


66  TRIGONOMETRY 

K=  ±ab.  ..*.  log  K=  log  a  +  log  6  -f  colog  2. 

773  log  2.88818  J  log  1.44409 
61  log  1.78533  J  log  0.89267 
356  log  2.55145 

^  SDO          o-  2  colog  9.69897  - 10 

FIG.  29.  K=  38652.7  log  4.58718 

Ex.  3.  By  use  of  four-place  tables  find  the  area  of  the 
right  triangle  in  which  A  =  37.32°  and  6=308  (see  Fig.  28). 

log  K  =  log  a  -f  log  308  +  colog  2. 
To  find  log  a,  a  =  308  tan  37.32°. 

308  log  2.4886 

37.32°  log  tan  9.8821 

a  log  2.3707 

308  log  2.4886 

2  colog  9.6990  -  10 
K=  36167  log  4.5583 

50.  Formulas  for  Area  of  a  Right  Triangle.  The  area  of  a 
right  triangle  may  often  be  obtained  more  readily  by  the  use 
of  a  formula  involving  only  the  particular  parts  of  the  triangle 
given.  Denoting  the  area  of  a  right  triangle  by  K,  let  the 
pupil  show  that 

When  the  two  legs  are  given,  K=  |-  db. 

When  an  acute  angle  and  the  hypotenuse  are  given, 

K  =  \  c*  sin  A  cos  A  (or  =  J  c2  sin  B  cos  B). 
When  the  hypotenuse  and  a  leg  are  given, 


c-a)  (or  = 

When  an  acute  angle  and  a  leg  are  given, 

lT  =  ltt2tan.B  (or  =  \  b2  tan  A) , 
or  K  =  ±a2cotA  (or  =  \  ¥  cot  B) . 

By  geometry,  what  is  the  method  or  formula  for  computing  the  area 
of  an  isosceles  triangle?  of  a  regular  polygon?  The  formulas  given  above 
for  computing  the  area  of  a  right  triangle  are  sometimes  useful  in  com- 
puting the  area  of  an  isosceles  triangle,  or  of  a  regular  polygon. 


RIGHT   TRIANGLES  67 

EXERCISE  21 

Using  five-place  tables,  compute  the  area  of  the  right  triangle  in 
which : 

1.  A  =  28°  18',  6  =  216.  5.   5  =  63°  18',  c  =  124.72. 

2.  .5  =  72°,  a  =  196.  6.    a  =  192.7,  b  =  212.97. 

3.  .4  =  21°  16' 30",   c  =  31.967.          7.   a  =  0.73216,  c=.9125. 

4.  c  =  46.72,  6  =  32.54.  8.    c  =  927.8  ft,  b  =  759.8  ft 
9.  Given  a  =  2.5  and  K=  4.27,  find  6,  c,  and  A 

10.  Given  K=  7.256  and  ^L  =  26°  18',  find  a,  b,  and  c. 

11.  Given  K  =  55.686  and  c  =  15.67,  find  a,  6,  and  A. 

Compute  the  area  of  the  isosceles  triangle  in  which : 

12.  Base  =  12.67,  leg  =  9.267. 

13.  Base  =  .67892,  altitude  =  .26217. 

14.  Base  angle  =  68°  18',  leg  =  .2892. 

15.  Vertex  angle  =  105°  17',  altitude  =  13.67. 

16.  Vertex  angle  =  113°  18',  leg  25.6. 

17.  Given  area  =  16.72  and  base  =  6.37,  find  altitude,  leg,  and  base 
angle. 

18.  Given  area  =  .9273  and  base  angle  =  27°  18',  find  leg,  base,  and 
altitude. 

19.  Given  area  =  22.76  and  vertex  angle  =  117°  55',  find  leg,  base, 
and  altitude. 

20.  Find  the  area  of  the  regular  pentagon  whose  perimeter  is  3.35. 

21.  Find  the  area  of  the  regular  dodecagon  whose  apothem  is  1.7267. 

22.  Find  the  area  of  a  regular  heptagon  inscribed  in  a  circle  whose 
radius  is  0.7516. 

23.  Given  a  regular  octagon  whose  apothem  is  2.27 ;  find  the  differ- 
ence between  its  area  and  that  of  the  inscribed  circle. 

24.  Given  n  =  9  and  K  =  30,  find  r,  c,  and  R. 

25.  Given  n  —  11  and  K  =  35,  find  the  perimeter. 

26.  Given  n  =  5  and  K  =  37,  find  p  and  R. 

27.  If  n  denotes  the  number  of  sides,  R  the  radius,  and  C  the  cen- 
tral angle  of  any  regular  polygon,  prove  that  K=nR2  sin  ^  C  cos  ^  (7. 


68  TRIGONOMETRY 

Using  four-place  tables,  find  the  area  of  each  of  the  following  right 
triangles,  given: 

28.  A  =  34.6°,  a  =  67.8.  32.  b  =  8.42,  c  =  11.26. 

29.  B  =  84°,  a  =  100.  33.  B  =  39.24°,  c  =  23.68. 

30.  A  =  18.62°,  b  =  72.36.  34.     c  =  5000,  a  =  3000. 

31.  a  =  .16376,  b  =  .19762.  35.  A  =  47°,  a  =  .0087. 

Solve  the  following  right  triangles,  given: 

36.  6  =  6.37,  K=  26.38. 

37.  K  =1200,  .4  =  63.18°. 

38.  K  =  .  4962,  c  =  .  1635. 

Find  the  area  of  each  of  the  following  isosceles  triangles,  given : 

39.  Base  =  .7262,   leg  =  .5263. 

40.  Altitude  =  12.36,   leg  =  17.27. 

41.  Altitude  =  86.27,   base  =  111.63. 

42.  Base  angle  =  42.67°,   leg  =  17.43. 

43.  Vertex  angle  =  100.24°,   altitude  =  8.217. 

44.  Vertex  angle  =  78.32°,   leg  =  .6526. 

In  an  isosceles  triangle : 

45.  Given  area  =  192.67  and  base  =  43.64,  find  altitude,  leg,  and 
base  angle. 

46.  Given  area  =  0.7362  and  base  angle  =  37.43°,  find  leg,  base,  and 
altitude. 

47.  Given  area=  1367.8  and  vertex  angle  =  113.28°,  find  base,  leg, 
and  altitude. 

48.  Given  area  =  .1025,  and  leg  =  .4916,  find  the  base,  altitude,  and 
angle  at  the  base. 

49.  Find  the  area  of  a  regular  decagon  whose  perimeter  is  27.63. 

50.  Find  the  area  of  a  regular  pentagon  whose  apothem  is  .4782. 

51.  Find  the  area  of  a  regular  heptagon  inscribed  in  a  circle  whose 
radius  is  116.2. 

52.  Given  the  side  of  a  regular  octagon  as  5.33,  find  the  difference 
between  the  area  of  the  octagon  and  that  of  the  circumscribed  circle. 


RIGHT   TRIANGLES  69 

In  a  regular  polygon : 

53.  Given  n  =  1  and  K  =  14,  find  c,  r,  and  R. 

54.  Given  n  =  11  and  K  =  1000,  find  r,  c,  and  R. 

55.  Given  ?i  =  9  and  K  =  47,  find  7*,  c,  and  7?. 

56.  Given  n  =  14  and  K=  800,  find  the  perimeter. 

Without  the  use  of  the  tables,  find  the  area  of  each  of  the  following 
right  triangles,  given: 

57.  a  =  100  and  A  =  60°.  61.  a  =  80  and  c  =  160. 

58.  b  =  600  and  c  =  1200.  62.  b  =  40  and  c  =  40  V2. 

59.  a  =  26.3  and  6  =  21.2.  63.  c  =  4000  and  ^4  =  30°. 

60.  B  =  60°  and  a  =  90.  64.  A  =  45°,  6  =  120. 

Also  of  each  of  the  following  isosceles  triangles,  given : 

65.  Vertex  Z  =  120°,  leg  =  100.  67.   Leg  =  40,  altitude  =  20. 

66.  Base  Z  =  30°,  base  =  200.  68.   Vertex  Z  =  90°,  leg  =  400. 

EXERCISE  22.    APPLICATIONS 

Solve,  using  either  set  of  tables : 

1.  The  angle  of  elevation  (see  Art.  88)  of  the  top  of  a  cliff,  measured 
from  a  point  225  ft.  from  the  base,  is  60°.     How  high  is  the  cliff  ? 

2.  At  a  point  170  ft.  from  a  tower,  and  on  a  level  with  its  base, 
the  angle  of  elevation  of  the  top  of  the  tower  is  found  to  be  70°  18' 
[70.3°].     What  is  the  height  of  the  tower  ? 

3.  The  angle  of   elevation  of   the  sun  is  65°  30'  [65.5°]  and  the 
length   of   a  tree's   shadow,   on   a  level   plane,   is   52   ft.     Find   the 
height  of  the  tree. 

4.  If  the  Eiffel  Tower  is  984  ft.  high,  what  will  be  the  angle  of 
elevation  of  its  top,  when  viewed  at  a  distance  of  a  mile  ? 

5.  The  length  of  a  kite  string  is  700  ft.,  and  the  angle  of  eleva- 
tion of  the  kite  is  44°  36'  [44.6°].     Find  the  height  of  the  kite  suppos- 
ing the  kite  string  to  be  straight. 

6.  One  of  the  equal  sides  of  an  isosceles  triangle  is  62.8  ft.,  and 
one  of  the  equal  angles  is  52°  18'  36"  [52.31°].     Find  the  base,  altitude, 
and  area  of  the  triangle. 

7.  What  is  the  elevation  of  the  sun,  if  a  tree  82.6  ft.  high  casts 
a  shadow  105.8  ft.  long  on  a  horizontal  plane? 


70  TRIGONOMETRY 

8.   A  ladder,  25  ft.  long,  leans  against  a  house  and  reaches  to  a 

point  21.6  ft.  from  the  ground.      Find  the  angle  between  the  ladder 

and  the  house,  and  the  distance  the  foot  of  the  ladder  is  from  the  house. 

Why  are  we  able  to  solve  an  example  like  this   by  trigonometry 

when-  we  are  not  able  to  do  so  by  geometry  ? 

9.   The  Washington  Monument   is   555 
ft.  high.     How  far  apart  are  two  observers 
555      who  from  points  due  west  of  the  monument 
observe  its  angles  of  elevation  to  be  25°  and 
48°  17'  [48.28°]  respectively? 

10.  If  the  Grand  Canon  of  the  Colorado  is  5000  ft.  deep,  what  will 
be  the  angle  of  depression  of  the  river  flowing  through  it  when  viewed 
from  the  brink  of  the  canon  at  a  horizontal  distance  of  3  mi.  ? 

11.  If  a  hillside  has  a  slope  of  7°,  a  dam  10  ft.  high  will  force  the 
water  how  far  back  up  the  hillside? 

12.  A  tower  125  ft.  high  stands  on   the   bank   of  a  river.     The 
angle  subtended  by  the  tower  at  the  edge  of  the  opposite  bank  is  23°  31' 
[23.52°].     Find  the  width  of  the  river. 

13.  What  is  the  height  of  a  hill  if  its  angle  of  elevation  taken  at 
the  foot  of  the  hill  is  40°  18'  [40.3°]  and  if  this  angle  taken  150  yd. 
from  the  foot  of  the  hill  and  on  a  level  with  the  foot  is  28°  42'  [28.7°]  ? 

14.  From  the  summit  of  a  hill,  there  are  observed  two  consecutive 
milestones  on  a  straight  horizontal  road  running  from  the  base  of  the 
hill.     The  angles  of  depression  (see  Art.  88)  are  found  to  be  12°  and  7° 
respectively.     Find  the  height  of  the  hill. 

15.  A  valley  is  crossed  by  a  horizontal  bridge,  whose  length  is  I. 
The  sides  of  the  valley  make  angles  ra  and  n  with  the  plane  of  the 
horizon.     Show  that  the  height  of  the  bridge  above  the  bottom  of  the 

valley  is 

cotm  +  cotn 

16.  Upon  a  hill  overlooking  the  sea  stands   a  tower   70  ft.  high. 
From  a  ship  the  angle  of  elevation  of  the  base  and  top  of  the  tower 
are  respectively  15°  4'  [15.07°]  and  15°40'  [15.67°].     What  is  the  height 
of  the  hill  and  the  horizontal  distance  of  the  ship  from  the  tower  ? 

17.  Given : 

Z.  AKF=  Z  ARK=  Z  RTF=  90°. 

Z  KAR  =  60°  and  AR  =  12. 
Without  the  use  of  the  tables  find  the 
length   of   all   the   other   lines   in    the 
figure.  A    12    R 


RIGHT   TRIANGLES  71 

18.  A  boy  standing  m  feet  behind  and  opposite  the  middle  of  a 
football  goal,  sees  that  the  angle  of  elevation  of  the  nearer  crossbar  is 
A,  and  the  angle  of  elevation  of  the  crossbar  at  the  other  end  of  the 
field  is  C.     Prove  that  the  length  of  the  field  is  m  (tan  A  cot  C—  1). 

19.  A  railroad  embankment  is  7  ft.  high.     If  the  top  of  the  embank- 
ment is  8  ft.  wide  and  the  sides  slope  at  an  angle  of  43°,  what  will 
be  the  width  of  the  base  ? 

20.  If  the  Metropolitan  Life  Insurance  building  of  New  York  City 
is  700  ft.  high,  how  far  from  the  building  is  an  observer  when  the 
angle  of  elevation  of  the  top  of  the  building  is  7°  36'  [7.6°]  ? 

21.  A  man  standing  on  the  bank  of  a  river  observes  that  the  angle 
of  elevation  of  the  top  of  a  tree  on  the  opposite  bank  is  60° ;  when  he 
retires  50  m.  from  the  edge  of   the  river,  the  angle   of   elevation   is 
30°.     Without  the  use  of  the  tables  find  the  height  of  the  tree  and  the 
width  of  the  river.  s 

22.  Given:         ATP=6m.; 

Z7T=Z  ^=60°;  Z SRN  =  45°; 


and  RNTP  a  square. 

Without   the   use   of    the   tables   find   the 
lengths  of  KR,  PR,  RS,  ST,  SF,  and  TF. 

23.  A  tower  and  a  monument  stand  on  the  same  horizontal  plane. 
The  height  of  the  tower  is  35.6  m.  and  the  angles  of  depression  of 
the  top  and  base  of  the  monument,  as  observed  from  the  top  of  the 
tower,  are  respectively  5°  16'  48"  [5.28°]  and  8°  18'  30"  [8.3°].     How 
high  is  the  monument  ? 

24.  A  flagstaff  stands  on  the  roof  of  a  building.     From  a  point  A 
on  the  ground  the  angles  of  elevation  of  the  foot  and  the  top  of  the 
flagstaff   are  37°   and   46°,   respectively.      From   a  point   B,   250   ft. 
farther  off  and  in  line  with  A  and  the  base  of  the  building  immediately 
below  the  flagstaff,  the  angle  of  elevation  of  the  top  of  the  flagstaff  is 
27°  30'  [27.5°].     Find  the  length  of  the  flagstaff. 

25.  From  the  top  of  a  lighthouse,  150  ft.  above  the  sea  level,  the 
angle  of  depression  of  a  buoy  situated  between  the  lighthouse  and  the 
shore  was  62°  14'  [62.23°]  and  that  of  a  point  on  the  shore  in  a  straight 
line  with  the  buoy  was  12°  10'  [12.17°].     Find  the  distance,  in  feet,  of 
the  buoy  from  the  shore. 

26.  The  base  of  a  rectangle  is  50.62  and  its  diagonal  is  71.6.     Find 
the  altitude  of  the  rectangle  and  the  angle  which  the  diagonal  makes 
with  the  base. 


72  TRIGONOMETRY 

27.    Given : 

0.4  =  1, 


Express   AB,    OB,   BC,    OC   in   terms   of 
trigonometric  functions  of  x  and  y. 

28.  The  Singer  building  of  New  York 
City  is  612  ft.  high.  Make  up  some  problem  concerning  this  which 
can  be  solved  by  trigonometry. 

29.  The  diagonals  of  a  rhombus  are  42.28  and  30.58.     Find  the 
sides  and  angles. 

30.  Make  up  (or  collect)  as  many  different  examples  as  you   can 
showing  the  practical  uses  of  the  solution  of  right  triangles  by  trigo- 
nometry, each  example  being  distinct  from  the  rest  either  in  principle 
or  in  the  field  of  its  application. 

31.  Who  first,  and  at  what  date,  taught  the  trigonometric  solution 
of  triangles  in  the  same  general  way  as  is  done  at  present  ? 


CHAPTER   IV 
GONIOMETRY 

TRIGONOMETRIC   FUNCTIONS   OF  ANGLES  IN   GENERAL 

51.  Angles  greater  than  90°.     In  solving  oblique  triangles, 
angles  greater  than  90°  may  occur.     Hence  it  is  important 
to  learn  what  the  trigonometric  functions  of  an  obtuse  angle 
are.     Similarly  the  radius  of  a  rotating  wheel,  as  in  a  dynamo, 
generates  angles  greater  than  360°  and  by  successive  rota- 
tions generates  angles  unlimited  in  size. 

In  astronomy,  the  heavenly  bodies,  by  successive  rotations  about  an 
axis,  and  by  revolutions  in  an  orbit,  also  generate  angles  unlimited  in  size. 

Hence  a  general  method  is  needed  of  determining  the 
trigonometric  functions  of  angles  unlimited  in  size. 

52.  The  Four  Quadrants.     Definitions.     Let  AC  (Fig.  30) 
be  the  horizontal  diameter  of  a  circle  ABCD,  and  BD  the 
diameter  perpendicular  to  AC. 

Then  AOB,  BOC,  COD,  and  DOA 
are  termed  the  first,  second,  third,  and 
fourth  quadrants  of  the  circle. 

On  Fig.  31  the  four  parts  into  which  a 
plane  is  divided  by  the  lines  XX'  and  TY1 
are  also  termed  quadrants  and  are  numbered 
in  the  same  order  as  the  quadrants  of  a 
circle. 

In  treating  of  the  properties  of  angles  in  general,  it  is 
convenient,  wherever  possible,  to  let  the  angles  start  at  the 
same  place,  as  OA  (that  is,  to  have  the  vertex  and  a  side  in 
common). 

Let  the  rotating  radius  start  in  the  position  OA  and  rotate 
toward  the  position  OB  (in  the  direction  contrary  to  that  in 
which  the  hands  of  a  clock  move,  or  counter-clockwise). 

73 


74 


TRIGONOMETRY 


The  AAOP19  AOP2,  AOP8,  AOP4  are  called  angles  in  the 
first,  second,  third,  and  fourth  quadrants  respectively. 

The  initial  line  of  an  angle  is  the  rotating  radius,  which 
generates  the  angle,  in  its  first  position,  as  AO. 

The  terminal  line  of  an  angle  is  the  rotating  radius  in  its 
final  position,  as  OP2  for  z  AOP%. 

By  continuing  the  rotation  of  OA,  angles  greater  than 
360°  will  be  generated.  If  two  angles  differ  by  360°,  or  by 
any  exact  multiple  of  360°,  they  will  have  the  same  terminal 
line. 

Coterminal  angles  are  angles  which  have  the  same  termi- 
nal line,  as  37°,  397°,  and  757°. 

In  general  an  angle  is  said  to  be  of  or  in  that  quadrant  in 
which  its  terminal  line  lies. 

53.  Negative  Angles.     In  algebra  it  is  shown  that  negative 
quantity  is  quantity  exactly  opposite  in  some  respect,  as,  for 
instance,  in  direction,  from  other  quantity  taken  as  positive. 
Hence  if  the   rotating   radius   move  from  the  position  OA 
(Fig.   30)   toward  the   position  OD    (that   is,   in  the  same 
direction  with  the  hands  of  a  clock,  or  clockwise),  a  nega- 
tive angle,  as  the  acute  Z  AOP4,  will  be  generated.     If  the 
radius  continue  to  rotate  in  this  direction,  a  whole  series  of 
negative  angles  will  be  formed  similarly. 

54.  Rectangular    Coordinates.      In    order    to    define   the 

trigonometric  functions  of  angles 
greater  than  90°,  and  of  nega- 
tive angles,  two  straight  lines, 
XX'  and  YY'  (Fig.  31),  inter- 
secting at  the  point  0  and  per- 
pendicular to  each  other,  are 
taken  and  called  axes.  The 
signs  of  other  lines  used  are  de- 
FiG.si.  termined  by  their  position  with 


GONIOMETRY 


75 


reference  to  these  axes  Lines  drawn  from  YY'  to  the  right 
(and  ||  XX')  are  taken  as  +  ;  lines  drawn  from  YY'  to  the 
left  (and  II  XX')  are  taken  as  - .  Lines  drawn  from  XX' 
above  (and  II  YY')  are  taken  as  +  ;  lines  drawn  from 
XX'  below  (and  II  YY')  are  taken  as  -. 

The  origin  is  the  point  in  which  the  axes  intersect,  as  the 
point  0  on  Fig.  31. 

The  ordinate  of  a  point  is  the  distance  of  the  point 
above  or  below  the  axis  XX'.  The  abscissa  of  a  point  is 
the  distance  of  the  point  to  the  right  or  left  of  the  YY'  axis. 
Thus,  the  ordinate  of  Pl  is  Mfi ;  the  abscissa  of  Pl  is  OM^ 

Coordinates  is  the  general  term  for  abscissa  and  ordinate 
of  a  point..  The  coordinates  of  a  point  may  be  written  to- 
gether in  parenthesis  with  abscissa  first  and  a  comma  be- 
tween. Thus  if  OMt  =  a,  and  Mfi  =  6,  the  coordinates  of 
Pl  are  (a,  6). 

The  distance  of  a  point  is  the  line  drawn  from  the  origin 
to  the  point,  thus  on  Fig.  31  the  distance  of  P1  is  OPi-  The 
distance  of  a  point  is  independent  of  sign. 

55.    Definitions  of  Trigonometric  Functions  of  Any  Angle. 


Y 

Y 

Y 

Y 

^\     f^XT 

\                  / 

Tx 

-N 

S\        \J\f\               N, 

M3: 

/' 

\M< 

0 

X    M2         0 

X 

V 

0           X         \        0 

>\ 

1 

/ 

3 

^4 

FIG.  32. 


FIG.  33. 


FIG.  34. 


FIG.  35. 


If  we  regard  an  angle  as  formed  by  an  initial  line  and  a 
line  drawn  from  the  origin  to  a  point  whose  abscissa  and 
ordinate  are  considered,  then 

sme  of  an  angle  =  ratio  of  ordinate  to  distance; 
cosine  of  an  angle  =  ratio  of  abscissa  to  distance; 


76 


TRIGONOMETRY 


tangent  of  an  angle  =  ratio  of  ordinate  to  abscissa 

cotangent  of  an  angle  =  ratio  of  abscissa  to  ordinate; 

secant  of  an  angle  =  ratio  of  distance  to  abscissa; 

cosecant  of  an  angle  —  ratio  of  distance  to  ordinate. 

Thus   in   Figs.  32,  33,  34,  35,  sin  z  XOP, 


issa;  ^^ 


sin  Z  XOP*  =          ,  sin  Z  XOP,  = 


,  sin  Z  ZOP4  = 

Let  the  pupil  point  out  in  like  manner  the  other  trigo- 
nometric functions  of  the  angles  XOP^  XOP2,  XOP3,  XOP*. 

56.    Trigonometric  Functions  represented  by  Lines. 

If  a  circle  (Fig.  36)  be  drawn  with  0  as  a  center  and  a 
radius  OA,  equal  to  1,  and  with  Mf^  M2P2,  M3P3,  M4P4, 
perpendicular  to  XX', 


]A     X 


A    x 


Similarly,  sin  Z  AOP2  =  M2P2 ;  sin  Z  AOP8  =  M3P3 ;  and 
sin  ^  J.OP4  =  M4P4.  ,  Or,  in  the  circle  as  described,  the'  sine 
of  an  angle  is  represented  by  a  line  drawn  from  the  terminal 
end  of  the  arc  intercepted  by  the  angle,  and  perpendicular  to  the 
horizontal  diameter. 


GONIOMETRY 


77 


Similarly  if  (in  Fig.  37)  Nfl9  N2P2,  N3Pv  N4P4  are  perpen- 
dicular to  YY',  i 

cos  Z  AOP,  =      &  =  Ml-NiPi  ; 


cos 


cos 


Or,  in  the  circle  as  described,  the  cosine  of  an  angle  is 
represented  by  a  line  drawn  from  the  terminal  end  of  the  arc 
intercepted  by  the  angle,  and  perpendicular  to  the  vertical 
diameter. 

Similarly  (in  Fig.  38),  if  TT'  is  tangent  to  the  circle  at  A, 


tan  Z.AOP*  =  AT2  ;    tan  Z  AOP3  =  ATZ  ;   tan  Z  AOP4  =  'AT*. 

Or  in  the  circle  as  described,  the  tangent  of  an  angle  is  repre- 
sented by  a  line  drawn  touching  the  initial  end  of  the  arc  inter- 
cepted by  the  angle,  and  terminated  by  the  radius  to  the  other 
end  of  the  arc,  produced. 


Rt R, 


FIG.  38. 


FIG.  39. 


Similarly  (in  Fig.  39),  if  R^  is  tangent  to  the  circle  at 
the  point  B, 

cot  Z^OP!  =  tan  z.BORl  =  ^  =  ^  =  BRl\ 

OB 

cot  ^AOP2  =  BR* ;  cot  z.AOP3  =  BRS ;  cot  Z.AOP±  =  BR± ; 
or  in  the  circle  as  described  the  cotangent  of  an  angle  is  repre- 


78 


TRIGONOMETRY 


sented  by  a  line  which  is  the  tangent  of  the  complement  of  the 
given  angle- 
On  Fig.  38  the  secants  of  the  four  angles  used  are  readily  shown  to 
be  represented  by  07\,  OT.2,  OT3,  OT4;  or,  in  general,  the  secant  of  an 
angle  is  represented  by  a  line  drawn  from  the  center  through  the  terminal 
end  of  the  arc  intercepted  by  the  angle,  and  terminated  by  the  tangent. 

Similarly  on  Fig.  39  the  cosecants  of  the  four  angles  used  are  repre- 
sented by  ORu  OR2)  OR3)  OR4;  or,  in  general,  the  cosecant  of  an  angle 
is  represented  by  a  line  which  is  the  secant  of  the  complement  of  the  angle. 

It  will  be  convenient  to  draw  a  figure  for  an  angle  in  each 
quadrant  showing  the  lines  which  represent  the  functions  of 
that  angle. 

R B 


The  lines  which  represent  the  various  trigonometric  func- 
tions of  an  angle  are  not  the  same  as  the  trigonometric 
functions  which  they  represent,  but  they  have  many  of  the 
game  properties  as  the  functions  or  ratios.  It  is  often 


GONIOMETRY  79 

easier  to  perceive  these  properties  by  the  use  of  the  lines, 
than  by  the  use  of  the  ratios  which  the  lines  represent. 

In  deriving  the  properties  of  the  trigonometric  functions 
of  angles  greater  than  90°  we  shall  derive  them  from  the 
lines  representing  the  functions ;  but  in  such  cases  we  give 
some  specimen  proofs  showing  how  these  properties  may 
be  derived  from  the  ratio  definitions  (of  Art.  55),  and  in  other 
cases  leave  it  as  an  exercise  for  the  pupil  to  derive  the  proofs 
from  the  ratios  if  the  teacher  considers  it  desirable. 

57.  Signs  of  the  Trigonometric  Functions  in  the  Different 
Quadrants.  Of  the  lines  representing  the  sines  of  angles  in 
the  different  quadrants,  viz.  M^P^  M<>P^  M3PS,  Mf± 
(Fig.  36),  the  first  two  are  above  the  horizontal  axis,  and  are 
therefore  plus  in  sign;  the  last  two  are  below,  and  therefore 
minus.  Hence  the  signs  of  the  sines  of  angles  in  the  four 
quadrants  are  respectively  + ,  + ,  — ,  — . 

The  students  may  obtain  the  same  results  from  Figs.  32-35  by  using 
the  general  definitions  of  trigonometric  functions  given  in  Art.  55. 


Similarly  in  Fig.  37  the  cosine  lines  NVP^  -ZV"2P2? 
N4P4  are  +,  -,  -,  +,  respectively;  and  in  Fig.  38  the 
tangent  lines  AT,,  AT,,  AT3,  AT±  are  +  ,  -,  +  ,  -, 
respectively. 

Since  the  sine  of  a  quantity  and  of  its  reciprocal  must  be 
the  same,  the  sign  of  the  cotangent  in  the  various  quadrants 
must  be  the  same  as  that  of  the  tangent ;  that  of  the  secant, 
the  same  as  the  cosine ;  that  of  the  cosecant,  the  same  as  the 
sine. 


Or,  proceeding  geometrically,  on  Fig.  39,  the  cotangent  lines 
BR2,  BE,,  BEA  are  +,  -,  +,  -. 

The  secant  is  considered  as  plus  when  it  is  drawn  in  the  same 
direction  from  the  center  as  the  terminal  radius  (thus  OT2,  Fig.  38,  is 
opposite  in  direction  from  OP2  and  is  therefore  negative).  Hence  the 
secant  lines  OTly  OT2)  OT3  OT4  have  the  signs  +,  — ,  — ,  +,  respec- 


80 


TRIGONOMETRY 


lively.     Similarly  the  cosecant  lines  (Fig.  39)   ORl}  OR2,  OR3,  OR+ 
have  the  signs  -f>   +>  — j   — - 

The  results  thus  obtained  may  be  arranged  in  a  table  as 
follows : 


I 

II 

Ill 

IV 

sine  and  cosecant 

+ 

+ 

- 

- 

cosine  and  secant 

+ 

- 

- 

+ 

tangent  and  cotangent 

+ 

- 

-+- 

- 

EXERCISE  23 
In  which  quadrant  is  each  of  the  following  angles  ? 

1.  123°.    1  6.   415°.    1  11.   1111°. 

2.  155°.    %  7.    - 18°.  U  12.    -  222°. 

3.  215°.    3  8.    -125°.?  13.     -1826°. 

4.  285°.    If  9.   612°.   )  14.   2625°. 

5.  338°.    /j  10.    -500°.  15.     -1500°. 

16.  Find  the  signs  of  the  functions  of  the  angles  in  Exs.  1,  3,  and  5. 

Give  two  positive  and  two  negative  angles  each  of  which  is  co- 
terminal  with : 

17.  25°.  18.    -30°.  19.    100°.  20.     -100°. 
Find  the  smallest  possible  angle  coterminal  with : 


21.  425°. 

22.  780°. 


23.  -300°. 

24.  875°. 


25.  -1760°. 

26.  1493°. 


In  which  quadrant  does  an  angle  lie : 

27.  If  its  sin  is  positive  and  cos  negative  ? 

28.  If  its  tan  is  positive  and  sin  negative  ? 

29.  If  its  cot  is  negative  and  cos  negative  ? 

30.  If  its  esc  is  negative  and  cot  positive  ? 

31.  If  its  cos  is  positive  and  tan  negative  ? 

32.  If  its  sec  is  negative  and  tan  negative  ? 

33.  A  railroad  embankment  is  9  ft.  high  and  43  ft.  wide  at  the  base. 
If  each  of  its  sides  makes  an  angle  of  27°  15'  [27.25°]  with  the  horizon- 
tal, how  wide  is  the  top  of  the  embankment  ? 


GONIOMETRY 


81 


FIG.  44. 


34.  If  a  railroad  embankment  is  7  ft.  high  and  28  ft.  9  in.  wide  at 
the  top,  and  one  side  has  a  slope  of  23°  30'  [23.5°]  and  the  other  a  slope 
of  32°  45'  [32.75°],  how  wide  is  the  base  ? 

35.  Make  up  a  similar  example  for  yourself. 

58.  Functions  of  0°,  90°,  180°,  270°,  360°.  In  Arts.  34  and 
35  it  is  shown  that  sin  0°  =  0  and  sin  90°  =  1.  Similar  results 
are  readily  perceived  for  other  quadrants  by  the  use  of  a  figure 
showing  the  sines  as  lines  in  the  different  quadrants. 

Thus  in  Fig.  44  in  the  first  quadrant 
the  sine  increases  from  0  to  1 ;  in  the 
second  quadrant  it  decreases  from  1  to  0 ; 
in  the  third  it  decreases  from  0  to  - 1 ; 
in  the  fourth  quadrant  it  increases  from 

-1  to  0.  Hence  the  sines  of  0°,  90°, 
180°,  270°,  360°,  in  order,  are  0,  1,  0, 

- 1,  0.  Similarly  in  the  first  quadrant 
(Fig.  45)  the  cosine  decreases  from  1  to  0  ; 
in  the  second  quadrant  it  decreases  from 
0  to  —  1 ;  in  the  third  quadrant  it  increases 
from  --  1  to  0  ;•  in  the  fourth  quadrant  it 
increases  from  0  to  1.  Hence  the  cosines 
of  0°,  90°,  180°,  270°,  360°,  in  order,  are 
1,0, -1,0,1. 

Similarly  from  Fig.  38,  or  from  the  formula  tan#=  sm  x .  it  is  clear 

cos  a; 

that  the  tangent  in  the  different  quadrants  changes  from  0  to  <x> ; 
from  —  oo  to  0 ;  from  0  to  oo  ;  from  —  oo  to  0.  Hence  the  tangents 
of  0°,  90°,  180°,  270°,  360°,  in  Order,  are  0,  ±  oc,  0,  ±  oo,  0. 

The  changes  in  the  value  of  the  cotangent,  the  secant,  and  the 
cosecant,  and  the  values  of  these  functions  for  the  above-mentioned 
angles  may  be  obtained  from  geometrical  figures  in  like  manner,  but 
these  values  are  obtained  more  readily  from  the  reciprocal  formulas 

cot  =  — •;  sec  = — •;  csc  =  ^-« 
tan  cos  sm 


Thus, 


sec  180°  = 


cos  180°      - 1 


82 


TRIGONOMETRY 


Obtaining  the  values  of  the  required  functions  thus  and 
arranging  all  the  results  obtained  in  a  table,  we  have 


0° 

90° 

180° 

270° 

360° 

sin 

0 

1 

0 

-1 

0 

cos 

1 

0 

-1 

0 

1 

tan 

0 

CO 

0 

00 

0 

cot 

00 

0 

00 

0 

00 

sec 

1 

00 

-1 

00 

1 

CSC 

00 

1 

00 

-1 

OD 

In  the  above  table  co  is  to  be  taken  as  +  or  --  according 
to  the  side  from  which  it  is  approached  (see  Art.  57). 

EXERCISE  24 

Find  the  numerical  value  of : 

1.  5  sin  90°  -f  7  cos  180°  +  8  sin  30°. 

2.  m  sin  0°  +  p  cos  90°  -f  c  cot  360°. 

3.  b  cos  90°  -  c  tan  180°  +  b  cot  270°. 

4.  (a2  _  C2)  cos  180o  +  4  ac  sin  90of 

5.  2  tan  0°  sin  90°  -  4  sec  0°  sin  270°  -f  5  esc  90°  cos  0°  cot  270°. 

6.  a  cos  180°  sec  360°-  b  tan  180°  sin  270°-  a  sin  90°  sec  0°  +  b  sin  90° 
cos  270°. 

7.  m  sin  270°  esc  90°  +  n  cos  180°  esc  270°  cot  270°  -  m  sec  180°. 

8.  6  m  esc  90°  cos2  0°  -  17  n  sec2  0°  cot2  270°  +  3  m  sin  270°  sec  360°. 

9.  Show  that 

4  cos2  45°  sec  0°  +  6  tan2  30°  sin  270°  + 12  cot2  45°  cos  180° 

-4  tan2  45°  esc  270°  =  -8. 

59.  Trigonometric  Functions  of  Angles  greater  than  360°. 
It  is  evident  that  the  trigonometric  functions  of  angles  from 
360°  to  720°  are  the  same  in  order  as  those  from  0°  to  360°. 
Similarly  for  every  succeeding  360°,  the  functions  repeat 
themselves. 

Hence  to  find  the  functions  of  an  angle  greater  than  360°, 
Divide  the-  angle  by   360°  and  find  the  required  trigono- 
metric function  of  the  remainder. 


GONTOMETRY  83 

Ex.    Sin  766°  =  sin  (2  x  360°  +  46°)  =  sin  46°. 

60.    Formulas  for  the  Acute  Angle  extended  to  any  Angle. 

The  equations  and  formulas  proved  in  Arts.  27-29  concern- 
ing the  function  of  an  acute  angle  are  true  for  the  functions 
of  any  angle. 

Thus,  on  each  of  the  Figs.  40-43,  MP2  +  OM2  =  OP\ 

That  is,  sin2  x  +  cos2  x  =  1. 

Also  in  each  quadrant  the  A  OMP,  OAT,  OBR  are  simi- 
lar. 

.'.  AT:  OA  =  HP  :  OM,  or  tanx:  1  =  sinx:  cosx, 

sinx 

or         tan  x  =  -    — . 

cosx 

» 

Let  the  pupil  prove  in  like  manner, 

1  1 

sin  x  =  -    — ,  cos  x  = 


esc  x  sec  x 


Or  these  results  may  be  proved  directly  from  the  ratio  definitions  of 
the  trigonometric  functions  of  any  angle. 

For  if  angle  XOP  of  Figs.  32-35  be  denoted  by  x,  in  any  quadrant 


abs.  P  +  ord.  P  =  dist.  P  , 

/abs.  P  V  ,  /ord.  P  V=  -i 
'  P)      (dist.  PJ 


Hence,  sin2  x  +  cos2  x  =  1. 

Let  the  pupil  prove  in  a  similar  manner  that 

tan2  x  -f-  1  =  sec2  x,  and  cot2  x  -f-  1  =  esc2  x. 

ord.  P 
Also         ta 


,  . 

abs.  P     abs.  P     cos  x  cos  x 

dist.  P 

Also    ord'-Pydist.  P=1     abs.  P      dist.  P=1   ord.  P     abs.  P=1  . 
'  dist.  P      ord.  P  ~    '  dist.  P      abs.  P        '  abs.  P     ord.  P~ 

or  sin  x  x  csc  x  =  1,  cos  x  x  sec  x  =  1,  tan  x  x  cotx=  1. 


84 


TRIGONOMETRY 


61.  One  function  of  an  angle  being  given,  the  other  functions 
may  be  found  in  a  manner  similar  to  that  used  in  Art.  30. 
Owing  to  the  fact  that  for  angles  less  than  360°,  two  angles 
correspond  to  any  given  function,  two  sets  of  answers  are 
found  in  each  example. 

Ex.  1 .    Given  cos  x  =  —  -|,  find  the  other  functions  of  x. 

By  the  table  of  signs  (Art.  57)  a  negative  cosine  occurs  in  both  the 
second  and  third  quadrants. 

2d  quadrant.       sin  x  =  Vl  —  (f  )2  =  Vl  —  |-f  =  V^V  =  t> 

ci  ti    nf*. 

=  —  f ,  etc. 

COS  X 

3d  quadrant. 


COS  X 

sin  x  =  Vl  —  (|)2  = 
tan  x  = 


-  =  — I  =  },  etc. 

COS  X         —  4 
<> 

Ex.  2.    Given  tan  x  =  2,  find  the  remaining  functions  of  x. 

The   positive   tangent  occurs   (see  Art.  57)  in  both  the  first  and 
third  quadrants. 

1st  quadrant,      sec2  x  =  1  -j-  tan2  x  =  1  -f-  4  =  5,  sec  x  =  V5, 

cos  x  = = =  -  V5,  etc. 

sec  x      V5     ^ 

3d  quadrant,      sec2  x  =  1  +  4,  sec  x  =  —  V5, 

cos  x  = = V5,  etc. 

-V5          5 

In  case  solutions  are  sought  by  the  geometrical  method,  the  follow- 
ing figures  may  be  used  in  Exs.  1  and  2  respectively. 


p 


-4 


PI 


FIG.  46. 


F 

FIG.  47. 


85 


EXERCISE  25 


1.  Find  the  numerical  value  of  sin  390° ;  also  of  cos  390°,  tan  390°, 
and  sec  390°. 

2.  Find  the  numerical  value  of  cos  780° ;  also  of  tan  780°,  sin  780°, 
and  cot  780°. 

3.  Find  the  values  of  sin,  cos,  tan,  and  cot  of  the  following  angles : 

4.  I8600.  6.    -675°.  8.    -1740°. 

5.  -330°.  7.    750°.  9.   2205°. 

10.  Given  cos  x  =  —  -| ,  find  the  other  functions  of  x. 

11.  Given  tan  x  =  —  -y,  find  the  other  functions  of  x. 

12.  Given  sin  x  =  —  -&,  find  the  other  functions  of  x. 

13.  Given  cot  x  =  2  and  sin  x  negative,  find  the  other  functions  of  x. 

14.  Given  sec  x  =  —  m  and  tan  x  negative,  find  the  other  functions 
of  x. 

15.  Given  tan  x  =  —  3,  find  the  other  functions  of  x  when  x  is  an 
angle  in  the  fourth  quadrant. 

16.  Given  sec  x  =  —  6,  find  the  other  functions  of  x  if  tan  x  is  posi- 
tive. 

17.  Verify  geometrically  the  results  obtained  in  Exs.  10-16. 

18.  Given  cot  y  =  f  V5  and  cos  y  negative,  find  sin  y  and  esc  y. 

19.  Given  tan  x  =  —  ^ V3  and  cos  x  positive,  find  the  other  func- 
tions of  x.  % 

20.  If  6  is  in  the  second  quadrant  and  if  cosec  0  =  -1/,  find  the  value 
P  cot  0  4-  sec  0 

tan  6  -|-  cos  0 

21.  Find  the  value  of  CQS^  +  cot^,  if  0  is  in  the  fourth  quadrant 

esc  0  H-  sec  0 
arid  tan  0  =  —  ^-. 

62.  Trigonometric  Functions  of  90° +  o?  in  terms  of  func- 
tions of  x.  The  trigonometric  functions  of  90°  +  xmay  be 
reduced  to  functions  of  x  by  use  of  the  following  formulas : 

sin  (90°  +  as)  =  cos  x.  cot  (90°  -f  x)  =  -  tan  x. 

cos  (90° +  x)  =  -  sin  x.  sec  (90°  +  x)  =  -  esc  x. 

tan  (90°  +  ac)  =  —  cot  x.  esc  (90°  +  a?)  =  sec  x. 


86 


TRIGONOMETRY 


For,   let   Z  AOP    (Fig   48  a)   be   any 
angle  x  in  the  first  quadrant.      Let  POQ 
P   be  a  right  angle.     Let  OP  =  OQ  =  1. 

Then 

/.  A 
.*.  sin 


(sides  _L) 
=  A  MOP.     (%p.  cmd  acute  Z  = ) 


FIG.  48  a. 


=  0#=  -PM=  -sins. 


tan  (90°  +  X)  =  sin  ;9Q°n°o+  *j  =  -°°^L  =  -  cot  x. 
cos90°  +  j       -smx 


-smx 

Let  the  pupil  supply  the  proofs  for  cot  (90°  +  x),  sec  (90°  +  x)> 
and  esc  (90°  4-  x). 

The  same  results  may  readily  be  obtained  for  angles  end- 
ing in  the  second,  third,  and  fourth  quadrants  by  use  of 
the  following  diagrams. 


Q 


FIG.  *48  6.  FIG.  48  c.  FIG.  48  d. 

Ex.  1.     Find  the  value  of  sin  300°. 

sin  300°  =  sin  (90°  +  210°)  =  cos  210° 

=  -  sin  120°  =  -  cos  30°  =  —  JV3. 

Ex.  2.     Reduce  tan  923°  to  a  function  of  an  angle  less 
than  90°. 

tan  923°  =  tan  (720°  +  203°)  =  tan  203°  (Art.  59) 

=  -  cot  113°  =  tan  23°. 

Ex.  3.     Simplify  cos  (630°  +  4). 

cos  (630°  +  A)'=  cos  (270°  +  A) 

=  -  sin  (180°  4-  A) 

=  -  cos  (90°  +  A)  =  sin  A. 


GONIOMETRY  87 

.     EXERCISE    26 

Find  the  numerical  value  of : 

1.  sin  210°.  4.    cot  150°.  7.   tan  210°. 

2.  cos  300°.  5.    sec  1215°.  8.    sin  330°. 

3.  tan  120°.  6.    sec  900°.  9.   cos  240°. 

10.  cos  225° +3  sin  330° -tan  225°. 

11.  cot 840°-3  tan  420°+2  sec  480°. 

Express  each  of  the  following  trigonometric  ratios  in  terms  of  a 
ratio  of  some  positive  angle  not  greater  than  45° : 

12.  sin  142°.  18.  cos  110°.  24.  sin  (280°  16'). 

13.  tan  163°.  19.  sin  567°:  '  25.  cot  (2100°  17 f). 

14.  cos  310°.  20.  cot  1415°.  26.  esc  1325°. 

15.  sec  185°.  21.  esc  1200°.  27.  cos  82°. 

16.  cot  265°.  22.  cos  117°.  28.  tan  1060°. 

17.  tan  315°.  23.  tan  428°.  29.  tan  840°. 

30.  Prove  sin  330°  cos  390°  =  cos  570°  sin  510°. 

31.  Prove  tan  45°  sec  1080°  cos  570°  sin  510° 

-  sin  330°  tan  225°  cos  390D  =  0. 

32.  Find  the  value  of  6  sec2 1080°  tan2 135°  sin  1890° 

+  8  cot  45°  cos  1140°  +  esc  630°  tan  225°  cos  720°  sin  1830°. 

Simplify  the  following  expressions  : 

33.  5  sin  (90°  +  a;)  —  6  cos  (180°  +  x). 

34.  a  sin  (90°  +  a?)  +  b  cos  (270°  +  x)  -  c  tan  (180°  +  a?). 

35.  p  sin  (180°  +  x)  cos  (180°  +  x). 

36.  (a  +  b)  sin  (270°  +  x)-(a-b)  cos  (270°  +  a?). 

63.    Trigonometric    Functions  of  a  Negative    Angle.     The 

trigonometric  functions  of  a  negative  angle  may  be  converted 

into  functions  of  a  positive  angle  by  use  of  the  following 
formulas : 

sin  ( —  x)  =  —  sin  x.  cot  ( —  x)  =  —  cot  x. 

cos  ( —  x)  =  cos  w.  sec  (—  x)  =  sec  x. 

tan  ( -  x)  =  -  tan  x.  esc  (—»)  =  —  esc  x. 


88 


TRIGONOMETRY 


For   let   /-  A  OP   (Fig.  49)  be   a   positive    angle,  x,  and 
AOQ  an  equal  negative  angle.     Let  OP  =  OQ  =  1. 
Then  the  right  triangles  0J/P  and  OMQ  are  equal. 
Hence, 

sin  ( —  x)  =  MQ  =  —  MP  —  —  sin  x 
cos  ( —  x)  =  OM  =  cos  x 

sin  ( —  x)       —  sin  x 


tan  ( —  x) 


cosx 


FIG.  49. 


•  cos  ( —  x) 
=  —  tan  x. 

Let  the  'pupil  supply  the  proofs  for 
cot  ( —  x),  sec  ( —  x),  and  esc  ( —  x). 

The  same  results  are  readily  obtained  for  angles  in  the 
other  quadrants  by  the  use  of  appropriate  diagrams. 

Ex.  1.    Find  the  numerical  value  of  cos  (—  225°). 
cos  (-  225°)  =  cos  225°, 

=  -  sin  135°  (Art.  62) 

=  —  cos  45°  =  —  -J-  V2,  Ans. 

Ex.  2.    Simplify  cot  (180°  -  A). 

cot  (180°  -  A)  =  -  tan  (90°  -  A), 

=  cot  (—  A)  =  —  cot  A,  Ans. 

64.  Reduction  Tables  and  General  Rules.  Some  of  the 
reductions  made  by  the  methods  of  the  preceding  articles 
are  usecl  so  frequently  that  it  is  convenient  to  collect  the 
results  obtained  by  them,  and  arrange  them  in  tables  for 
future  reference.  Thus 

sin  (90°  —  x)  —  cos  x. 

cos  (90°  -  x)  =  sinx. 

tan  (90°  -  x)  =  cot  x. 

cot  (90°  -  x)  =  tan  x. 


sin  (180°  —  x)  =  sinx. 
cos  (180°  •-  x)  =  —  cos  a:. 
tan  (180° -x)  =  -tan  a! 
cot  (180°-  x)  =  -cot a; 
sec  (180°  -x)  =  -seccc 
esc  (180°  —  x)  =  esc  x 


sec  (90°  -  x)  =  cscx. 
esc  (90°  —  x)  =  sec  x. 
Let  the  pupil  form    similar  tables  for  the   functions  of 
270°  -  x,  360°  -  x,  180°  +  x'9  270°  +  x. 


GONIOMETRY  89 

Or  the  following  general  rule  may  be  used : 
Each  function  of  18Q°±  a;  or  360°  ±  x  is  equal  in  absolute 
value  to  the  like-named  function  of  x;    but  each  function  of 
90°  ±  x  or  270°  ±  x  is  equal  in  absolute  value  to  the  co-named 
function  of  x.* 

For  example,  sin  (180°  +  x)  and  sin  x  by  the  above  rule  are  equal  in 
absolute  value.  But  it  must  also  be  remembered  that  they  are  opposite 
in  sign.  For  if,  for  instance,  x  is  acute,  180°  +  x  is  an  angle  in  the 
third  quadrant  and  therefore  sin  (180°  +  x)  is  negative.  But  x  mean- 
time'would  be  an  angle  in  the  first  quadrant,  hence  sin  x  would  be 
positive.  Hence,  in  general, 

sin  (180°  +  x)  =  -  sin  x. 

Let  the  pupil  show  in  like  manner  that,  by  the  above  rule, 
sin  (360°  —  x)  =  —  sin  x ;  also  that  sin  (270°— x)  =  —  cos  x. 

In  applying  the  above  general  rule  to  any  particular 
example  it  will  be  found  that  the  algebraic  sign  of  the  result  is 
the  same  as  the  sign  of  the  original  function. 

Thus,  sin  330°  =  sin  (360°  -  30°)  =  -  sin  30°,  the  short  way  of  deter- 
mining the  sign  of  sin  30°  being  to  note  that  sin  330°  is  negative  since 
330°  is  in  the  fourth  quadrant  and  that  sin  30°  must  have  the  same 
sign  as  sin  330°. 

If  geometrical  proofs  for  the  above  reduction  formulas  are  desired, 
such  proofs  may  be  obtained  by  following  the  methods  of  Art.  62.     But 
in  such  proofs,  when  constructing  an  angle 
like  180°  -|-  x,  or  270°  -f  x  on  the  diagram,  it 
is  an  advantage  to  construct   the  180°,  or 
270°  first,  beginning  with  the  initial  line,  and 
then  to  annex  the  angle  x  to  the  180°,  or 
270°,  after  it  has  been  constructed. 

Thus,  to  prove  that  tan  (270°  +  x)=  —  cot  x 
when  x  is  an  angle  in  the  second  quadrant 
(i.e.  an  obtuse  angle)  we  first  take  (Fig.  50) 
the  positive  angle  AOB'  (270°)  and  annex 
to  it  Z.B'OP  (=x  or  AAOP).  Then 

*  At  this  point  it  is  often  advantageous  to  have  the  class  study  the  solution  of 
Case  I  of  oblique-angled  triangles  (Arts.  74,  79).  This  shows  the  pupil  an 
important  application  of  the  preceding  principle  and  introduces  variety  into  the 
course  of  study. 


90 


TRIGONOMETRY 


x)=ZAOT    (as    indicated    by    the    long    bent    arrow),    and 
tan  (27Q°  +  x)  =  AT.     Also  cot  x  (or  cot  AOP)  =  BR. 

But  ZB'OT  =  ZAOR  (construction) 

Subtracting  90°  from  each  of  these  angles  we  have 

ZAOT  =  ZBOP.     .••„  AAOT  =  ABOP.      (leg  and  acute  Z  =) 
.-.  AT  =  .B.R,  in  absolute  magnitude.  (horn,  sides  of=  A) 

.-.  tan  (270°  +  x)  and  cot  a;  are  equal  in  absolute  magnitude. 

But  AT  and  BR  are  opposite  in  sign. 
.-.  tan  (270°  +  x)  =  —  cot  x. 

Similarly,  to  prove  sin  270°  —  x  =  —  cos  x 
when  x  is  an  angle  in  the  second  quadrant 
(Fig.  51)  we  take  Z  AOB'  (270°)  and  from 
it  deduct  Z  B' OP'  (=ZAOP  or  x).  Hence, 
sin  (270°  -  x)  =  MP1,  while  cos  x  =  JVP. 

Since  A  OMP'  =  A  O^P.  MP'  and  ^P  are 
equal  in  absolute  magnitude.  They  are  also 
opposite  in  sign. 

.°.  sin  (270°  —  x)  =  —  cos  x. 


EXERCISE  27 
Find  the  numerical  value  of : 


7.  sec  (-240°). 
a  tan  (-150°). 
9.  sin  (-135°). 


1.  sin  (-225°).  4.   cot  (-210°). 

2.  tan  (-300°).  5.   tan  (-600°). 

3.  cos  (-  120°).  6.    sin  (-  900°). 

Keduce  the  functions  of  the  following  negative  angles  to  the  functions 
of  positive  angles  not  greater  than  45° : 

10.  -119°.  13.     -15°.  16.    -900°. 

11.  -81°.  14.    -253°.  17.    -216°  43'. 

12.  -195°.  15.     -1000°.  18.    -307.24°. 

19.  Show  that  sin  420°  cos  390°  =  1  -  cos  (-  300°)  sin  (-  330°). 

20.  That  3  tan  (-  60°)  cot  (-  210°)  +  9  sin  (-  240°)  cos  (- 150°)  =  f . 

By  the  general  rule  stated  in  Art.  64  reduce  each  of  the  following  to 
a  function  of  x : 


21.  cos  (180°  +  x). 

22.  sin  (270°  +  #). 


23.  cos  (270°  -  x). 

24.  tan(180 


25.  sec  (180°  -  aj). 

26.  esc  (270°  +  a?). 


GONIOMETRY  91 

Simplify  the  following  expressions  : 

27.  5  sin  (90°  —  x)  +  8  cos  (180°-  x  ). 

28.  a  sin  (270°  -  x)  -  b  cos  (270°  —x)  +  c  tan  (180°  -  x). 

29.  m  cos  (180°  +  A)+p  cot  (180°  —  A)  +  q  tan  (270°  +  ^4). 

30.  sin  (270°  +  a;)  cos  (270°  -  x)  sin  (180°  -  a;). 

31.  sin  (aj  -  90°)  +  cot  (x  -  90°)  +  tan  (a;  -  180°). 

65.    General    Solutions    of    Trigonometric   Equations.      If 

there  be  no  limit  to  the  size  of  an  angle,  an  indefinite  num- 
ber of  angles  will  satisfy  every  trigonometric  equation  (see 
Art.  38). 

Ex.  1.    Solve  sin  x  =  \. 

There  are  two  angles  less  than  360°  whose  sine  is  |-,  viz.  :  30°  and 
150°.  If  360°,  or  any  multiple  of  360°,  be  added  to,  or  subtracted  from, 
each  of  these  angles,  the  sine  is  unchanged. 

Hence,  in  the  above  example,  x  =  30°  ±  n  (360°),  150°  ±  n  (360°), 
where  n  =  0  or  any  positive  integer. 

Ex.  2.    Solve  tana:-  ±V5- 


_ 
" 


60°    ±  n(360°),  120°  ±  n 


[240°  ±  n(360°),  300°  ±  w(360°). 
Ex.  3.   Solve  sin2x=cos2x. 

1  —  COS2  X  =  COS2  X. 


cos  x  =  ±  i  V2. 

=  {45°±  w(360°),  315°  ±  n(360°), 
~  |l35°±  »(360°),  225°  ±  »(360°). 
Or  more  briefly,     x  =  ±n  (180°)  ±  45°.     Ans. 

The  pupil  should  observe  that  the  values  of  a;  in  a  trigonometric 
equation  differ  in  an  important  respect  from  the  values  of  x  in  an 
algebraic  equation.  Thus,  in  an  algebraic  equation  the  values  of  x  are 
the  roots  of  the  equation  and  the  number  of  values  which  x  has 
equals  the  degree  of  the  given  equation.  Whereas,  for  instance  in 
Ex.  3  above,  the  roots  are  the  values  of  cos  a?,  while  the  values  of  x  are 
inferred  from  the  values  of  cos  x  and  may  be  unlimited  in  number 
no  matter  what  the  degree  of  the  original  trigonometric  equation. 


92  TRIGONOMETRY 

EXERCISE  28 

Solve,  the  following  trigonometrical  equations,  for  values  of  x  or  0. 

1.  sin  x  =  J.  10.  2  V3  cot  0  -  f  esc2  0=1. 

2.  cos2x  =  f.  11.  tan  0  -f  sec2  0  =  3. 

3.  tan2  a  =  l.  12.  cos2  0  +  cot2  0  =  3  sin2  0. 

4.  tan  x  =  i  cot  x.  13.  1  cot  0  —  cos  0  -f-  sin  0  =  1. 

5.  sin  x  +  esc  x  =  f .  14.  sec2  0  esc2  0  +  2  esc2  0  =  8. 

6.  tan2  x  —  sec  x  =  1.  15.  2  V3  tan  0  =  3  sec2  0  —  6. 

7.  2cos2a;-3sina;=0.  16.  4  sec20  -  7  tan20  =  3. 

8.  tan  x  +  cot  a  =  2.  17.  cot  0  +  2  tan  0  =  f  sec  0. 

9.  cot  x  +  esc2  a;  =  3.  18.  sin  0  +  V3  cos  0  =  2.. 

19.  A  ship  starting  from  a  certain  point  sailed  at  the  average  rate 
of  9.25  mi.  per  hour  on  a  course  22°  15'  [22.25°]  north  of  east.     At  the 
end  of  7  hr.  45  min.,  how  far  east  of  her  starting  point  would  she  be  ? 
How  far  north  ? 

20.  If  a  railroad  embankment  is  11  ft.  high,  76  ft.  wide  at  the  base, 
and  49  ft.  wide  at  the  top,  and  its  two  sides  have  the  same  slope,  find 
the  angle  at  which  each  side  slopes. 

21.  In  an  oblique  triangle  ABC,  A  =  127°  36'  [127.6°],  AB  =  472  ft., 
AC  =374:  ft.     By  dividing  the  triangle  into  right  triangles  and  solving, 
find  BC. 

22.  Pisa  spring  of  water,  Q  is  a  house,  and  R  is  a  barn.     If 
QR  =  217  ft.,  Z  PQR  =  63°  40'  [63.67°],  Z  PRQ  =  58°  15'  [58.25°],  find 
the  distance  of  the  spring  from  the  house  and  also  from  the  barn,  by 
solving  right  triangles  only. 


CHAPTER  V 


GONIOMETRY     (Continued) 


66.  Formulas  for  sin  (ae  + 1/)  and  cos(a?  +  ?/).  In  Fig.  52 
let  AOQ  be  an  angle  x,  and  QOP  an  angle  y,  the  sum 
of  x  and  ?/  being  less  than  a  right  angle. 

Let      OP  =  1.        Draw      PM  J_  OJ., 
P$_L  OQ,  QR^PM. 

Then   ZPPQ=Zx   (sides  -L), 

PQ  =  sin  ?/,  0$=  cos  ?/. 
sin  (x  4-  ?/)  =  PJf =  $JV"4-  PP. 
In  rt.  A  OQN,  QN=  sin  x  0$  (Art.  41)  =  sin  x  cos  y. 
In  rt.  A  RPQ,  PP  =  cos  xPQ  =  cos  x  sin  T/. 
H^nce,  sin  (a?  4-  */)  ?=  sin  a?  cos  y  4-  cos  as  sin  ?/. 
Also  on  Fig.  52,  cos(x  +  y)=  OM=  ON-RQ. 
In  rt.  A  OQN,  ON=  cos  x  OQ  =  cos  x  cos  y. 
In  rt.  APPQ,  PQ=sinxPQ=sinxsin2/. 
Hence,  cos  (ac  +  y)  =  cos  x  cos  y  —  sin  a?  sin  y. 

If  x  and  ?/  be  acute  angles  whose  sum  is  an  obtuse  angle,  the 
above  proofs  will  hold  good  without  any  change  except  that  it 
•E^       \          is  necessary  to  notice  that  in  the  statement 
cos  (x  +  y)  =  OM=  ON-  RQ,  OM  is  a  neg- 
ative line  and  is  obtained  by  subtracting 
M     o  N  the    positive   line   RQ   from   the  smaller 

FIG.  53.  positive  line  ON.     See  Fig.  53. 

If  either  x  or  y  is  obtuse,  the  above  formulas  may  be 
proved  as  follows : 


94 


TRIGONOMETRY 


Taking  x  and  y  as  still  acute, 

sin  (90°  +  x  +  y)  =  cos  (x  +  y)  (Art.  62) 

=  cos  x  cos  y  —  sin  x  sin  y. 

But  cos  B=  sin  (90°  +  x),-  sin  z  =  cos  (90°  +  z).      (Art.  62) 

/.  sin  (90°  +  x  4-  y)  =  sin  (90°  +  x)  cos  y  +  cos  (90°  +  x)  sin  ?/. 

Replacing  90°  4-  x  by  #', 

sin  (#'  +  y)  =  sin  x'  cos  y  +  cos  x  sin  ?/,  where  #'  is  an  obtuse 
angle. 

In  like  manner  the  formula  can  be  extended  to  the  case 
where  y  is  an  obtuse  angle.  The  formula  for  cos  (x  +  y)  may 
also  be  extended  in  like  manner. 

By  •  successive  additions  of  90°  to  x  and  y,  these  angles 
may  thus  be  made  any  angles  however  large.  In  like  manner 
the  formulas  may  be  shown  to  be  true  when  x  and  y  are 
diminished  by  any  integral  multiple  of  90°.  Hence,  the 
above  formulas  are  true  when  x  and  y  are  any  angles. 

Ex.    Taking  the  functions  of  30°,  45°,  60°  as  known,  find 

sin  75°. 

sin  75°  =  sin  (45°  +  30°)  =  sin  45°  cos  30°  +  cos  45°  sin  30° 


,  Am. 


67.    Formulas  for  sin  (w  —  y)  and  cos  (oc  —  y}.     In  Fig.  54 
let  AOQ  be  a  positive  acute  angle  x,  and  POQ  a  smaller 
angle  y,  subtracted  from  x. 


Let  OP=l-      draw   PM-LOA, 

PQA.  OQ,  QN±  OA,  PE±  QN. 

Then  Z.RQP=^x.  (sides  -L) 

Also          PQ  =  s'my,   OQ  =  cosy. 
sin  (x-y)  =  PM=  QN-  RQ. 


In  rt.  A  OQN,  QN=  sin  x  OQ  =  sin  x  cos  y. 


GONIOMETRY  95 

In  rt.  A  RQP,  RQ  =  cos  x  PQ  =  cos  x  sin  y. 
Hence,  sin  (x  —  y)  =  sin  w  cos  y  —  cos  oc  sin  y. 
Also  on  Fig.  54, 

cos  (x-y)  =  OM=  ON+  RP. 
In  rt.  A  OQN,  ON=  cos  x  OQ  =  cos  x  cos  y. 
In  rt.  A  RQP,  RP  =  sin  x  PQ  =  sin  x  sin  y. 
Hence,       cos  (a?  —  y)  =  cos  a?  cos  y  +  sin  ae  sin  t/. 

By  the  same  method  as  that  used  in  Art.  66  these  formulas 
can  be  proved  true  when  x  and  y  are  any  angles. 

Ex.    Obtain  the  numerical  value  of  cos  15°. 

cos  15°  =  cos  (45°  -  30°), 

=  cos  45°  cos  30°  +  sin  45°  sin  30° 


Ans. 

68.    Formulas  for  tan  (x  +  y)  and  tan  (x  —  y).    By  Art.  66, 

sin  (x  +  y)      sin  x  cos  y  +  cos  x  sin  y 
=—  —  s-£- 

cos  (#  4-  2/)     cos  x  cos  y  —  sin  a:  sin  y 

Divide  both  numerator  and  denominator  of  the  last  fraction 
by  cos  x  cos  y. 

sin  x  cds  y     cos  x  sin  ^/ 

m,  v        COS  X  COS  7/       COS  X  COS  ty 

Then,         tan  (x  +  y)  =  • 


cos>a^cos  y     sin  ^  sin 
cos  £  cos  i      cos  x  cos 


tan 
tan 


1  -  tan  x  tan  y 

Similarly,  let  the  pupil  show  that 

tan  x  -  tan  y 

tan  (oc-y}  =  —  —  , 

1  +  tan  x  tan  y 

.       COt  '05  COt  ?/  T  1 

and  cot  (x  ±  ;</)  = 


cot  y  ±  cot  a? 


9G  TRIGONOMETRY 

Ex.    Find  the  numerical  value  of  tan  105°. 

tan  105°  =  tan  (60°  +  45°) 

tan  60°  +  tan  45° 


-  tan  60°  tan  45 
V3  +  1 


1-V3-1      1-V3 

EXERCISE  29 

1.  If  sin  x  =  |-,  cos  x  =  f  ,  sin  y  =  -f^,  cos  y  =  j-f  ,  find  the  value  of 
sin  (x  +  y). 

2.  Also  of  sin  (x  —  y),  cos  (x  +  y),  and  cos  (x  —  y). 

3.  Find  sin  (a?  +  45°),  cos  (30°  —  a;),  and  sin  (x  —  60°)  in  terms  of 
sin  x  and  cos  x. 

4.  If  tan  x  =  i,  and  tan  y  =  2,  find  the  value  of  tan  (a?  -+-  y). 

5.  If  cot  x  —  —  2,  and  cot  y  =  -J-,  find  the  value  of  cot  (x  —  y). 

Find  the  numerical  value  of  : 

6.  cos  75°.  8.    sin  105°.  10.    sin  15°. 

7.  tan  75°.  9.   cot  105°.  11.    cos  105°. 

12.  Putting  90°  =  60°  +  30°,  find  sin  90°  ;  also  cos  90°. 

13.  State  in  general  language  the  formulas  proved  thus  far  in  this 
chapter  (thus  for  sin  (x  -f  y)  =  sin  x  cos  y  +  cos  x  sin  y,  say  "  the  sine 
of  the  sum  of  two  angles  equals  sine  of  the  1st  angle  times  cosine  of 
the  2d  plus  cosine  of  1st  times  sine  tif  2d  "). 

14.  Find  tan  (45°  -f  y),  and  also  tan  (45°  —  y)t  in  terms  of  tan  y. 

15.  Find  cot  (60°  +  y),  and  also  cot  (30°  +  y),  in  terms  of  cot  y. 

16.  Show  that  sin  (60°  -f-  45°)  +  cos  (60°  +  45°)  =  cos  45°. 

Prove  the  following  identities  : 
17. 


1  +  cot  A 

18.  cot(45°-^)  =  cot^  +  1. 

cot  A  -  1 

19.  sin  (60°  +  A)  —  sin  (60°  —  A)  =  sin  A. 

20.  cos  x  —  sin  x  =  V2  cos  (x  +  45°). 

21.  cos  x  -f  sin  x  =  V2  cos  (x  —  45°). 

22.  Find  the  smallest  value  of  x  which  will  satisfy  the  equation 

tan  (x  +  45°)  +  cot  (x  -  45°)  =  0. 


GONIOMETRY  97 

69.    Functions  of  the  Double  Angle.     In  the  formula 

sin  (x  +  y)  =  sin  x  cos  y  +  cos  x  sin  y, 
let  y  have  the  value  x ; 

then,  sin  (a;  +  a;)  =  sin  x  cos  a;  +  cos  x  sin  a: 

or,  sin  2  a»  =  2  sin  x  cos  oc. 

Similarly  from  the  formulas  for  cos  (a;  4-  y),  tan  (a:  +  y),  and 
cot  (a;  +  y),  let  the  pupil  obtain 

cos  2x  —  cos2  a;  —  sin2  x. 

2  tan  x 

l-tan2*? 

cot2a?-l 

2  cot  x 

Substituting  1  —  sin2  x  for  cos2  x  in  the  formula  for  cos  2  x9 

cos  2x  =  l  —  2  sin2  05. 
Substituting  1  —  cos2  a;  for  sin2  x  in  the  same  formula, 

cos  2x  =  2  cos2  x  —  1. 

Ex.    Find  cos  120°  from  the  functions  of  60°. 

cos  120°  =  cos  2  x  60° 

=  cos2  60°  -  sin2  60° 


EXERCISE  30 

1.  Given   sin  30°  =  £,   and   cos  30°  =  iV3,   find   sin  60°.       Also 
cos  GO0. 

2.  Given  tan  30°  =  |  V3,  find  tan  60°. 

3.  By  the  formulas  of  Art.  69,  find  the  value  of  sin  120°  and  tan  120°. 

Prove  the  following  identities  : 


4.   Sin2^=:  .  6. 


. 

1  -f  tan2  A  sin  x        cos  x 

2  4_l-tan2^  l  +  siii20_(tan0  +  l)2 

^-  l-sin20-(tan0-l)2' 


98  TRIGONOMETRY 

8.  State  the  formulas  for  sin  2  x  and  cos  2  x  in  general  language. 

9.  Find  sin  3  x  in  terms  of  sin  x. 

10.  Find  cos  3  x  in  terms  of  cos  x. 

11.  Find  tan  3  x  in  terms  of  tan  x. 

12.  Prove  sin  40  =  4  sin  0  cos  0  —  8  sin3  0  cos  0. 

13.  Given  tan  0  =  f ,  find  tan  2  0. 

14.  Given  cos  0  =  f ,  find  cot  2  0. 

In  a  right  triangle,  C  being  the  right  angle,  prove : 

15.  tan  B  =  cot  A. 

O  nh 

16.  tan  2^4  =  -J     — ,.  17.    sin  (A  -  B)  +  cos2^t  -  0. 

6-  —  a2 

18.  Show  that  sin2  x  =  i-™82*,  and  sin2  2x  =  1  ~  c°s  4  *. 

19.  Show  that  cos2  x  =  1  +  c;os2*,  and  cos2  2  x  =  l  +  <^4* 

2  2 

20.  Using  the  results  of  Exs.  18  and  19,  transform  sin4  x  into 

^  cos  4  #  —  i  cos  2  x  -f  -| . 

21.  Also  transform  cos4  x  into  an  expression  in  terms  of  cos  2  x  and 
cos  4  #. 

22.  Also  show  that  cos6  x  may  be  changed  to  the  form 

^e  (5  +  8  cos  2  x  —  2  sin2  2  x  cos  2  a  +  3  cos  4  a;). 

70.   Functions  of  the  Half  Angle. 

From  Art.  69,          cos  2  A  =  1  -  2  sin2  A. 
Hence,  2  sin2  A  =  l  -  cos  2  ^4. 

Let  A  =  \x\  then  2  J.  =  x. 

Hence,  2  sin2  \  x  =  1  —  cos  x. 


.       -,  .    1  -  COS 

.'.  sm  \w=  ±\ 


Similarly,  from        cos  2  J.  =  2  cos'2  ^4.  —  1, 


we  obtain,  cos     oc  =  ± 


GONIOMETRY  99 


A  T  -i        sn    x        . —  cos  x 

Also  tani£  = ?—  =±\- 

x  1  +  cosx 


1  +  cos  oc 

This  formula  may  be  reduced  to  another  convenient  form, 
thus : 

tan  ±x  =  V-      (1~COS:r)'2         =  J(l-cosx)?=l-co8g 
(1  +  cos  x)  (1  —  cos  x)          1  —  cos2  x          sin  x 


Similarly,  cot  *a?= 


sn  a? 

+  cos 
sin  a? 


Ex.    Find  tan  22^°  from  the  functions  of  45°. 

nooio      l-cos450_l-jV2      2-V2 

sin  45°          -~       -—  = 


EXERCISE  31 

1.  State  the  formulas  for  sin  \  A,  cos  1  A,  and  tan  ^  A  in  general 
language. 

2.  Given  cos  30°  =  1  V3,  find  sin  15°,  tan  15°,  cos  15°. 

Nj    3.    Given  sin  45°  =  i-V2,  find  cot  22^°,  cos  22|°,  sin  221°. 

4.  Given  cos  90°  =  0,  find  the  functions  of  45°. 

5.  Given  sin  A  =  f,  and  A  acute,  find  cos  1  A,  cot  1  A,  tan  -J  A 

/3  A  f\ 

6.  Given  cos  0  =  a,  find  cos  -,  cot  -,  tan  — 

222 

Prove  the  following  identities  : 

7.  tan    .1=      sin^    .  9. 


\ 


. 

1  +  cos  ^4  2      sec  ^  +  1 

2 


.8.   cot        =  -  .  10. 


-  .  .  . 

1  —  cos  J.  2      sec  ^  —  1 

11.  sin  \A  +  cos  -J-  J.  =  Vl  4-  sin  A 

12.  Express  cos  ^4,  sin  J.,  and  cot  ^4,  in  terms  of  cos  2  A 

13.  Find  the  value  of  tan?^  +  seca;  .f  ^  ig  in  the  second  quadrant 

cot  i  x  +  cos  aj 
and  sin  #=. 


100  TRIGONOMETRY 

14.    If  x  is  in  the  fourth  quadrant  and  esc  x  =  —  j,  find  the  numerical 
value  of  sin  frs+  sees 
cot    x  +  cosx 


15.  In  a  right  triangle  show  that  tan  \A  = 

*c  +  b 

16.  By  use  of  this  formula  solve  the  right  triangle  in  which  c  =  122 
and  a=120  (that  is,  the  Ex.  of  Art.  46). 

17.  If  the  diagonal  of  a  rectangle  is  171  in.  and  one  side  of  the 
rectangle  is  13  ft.  7  in.,  find  the  angle  between  the  diagonal  and  side. 

18.  Make  up  and  solve  a  similar  example  for  yourself. 

71.    Sum  or  Difference  of  Two  Sines  or  of  Two  Cosines  (Log- 
arithmic Formulas). 

Adding  and  subtracting  the  formulas  of  Art.  66,  and  also 
those  of  Art  67, 

sin  (x  +  y)  +  sin  (x  —  y)  =  2  sin  x,  cos  y      ...  (a) 

sin  (x  +  y)  —  sin  (x  -  y)  =  2  cos  x  sin  y      .     .     .  (b) 

cos  (x  +  y)  +  cos  (x  —  y)  =  2  cos  x  cos  y      .     .     .  (c) 

cos  (x  +  y)  —  cos  (x  —  y)  =  —  2  sin  x  sin  y  .     .     .  (d) 
If  we  let               x  +  y  =  A,  and  x  —  y=  B, 
then                         x  =  %(A  +  B),  and  y  =  %(A  -  B). 

Hence,  by  substitution  in  (a),  (b)9  (c),  (d), 

(1) 

(2) 
(3) 
(4) 


These  formulas  enable  us  to  convert  the  sum  or  difference 
of  two  sines,  and  also  of  two  cosines,  into  a  product  of  two 
functions,  and  hence  open  the  way  in  certain  examples  for  us 
to  save  labor  by  the  use  of  logarithms. 


Gt)NIOMETRY 

Ex.     Convert  sin  50°  +  sin  30°  into  a  product. 

By  formula  (1), 

sin  50°  +  sin  30°  =  2  sin  1(50°  +  30°)  cos  i(50°  -  30°) 
=  2  sin  40°  cos  10°. 


EXERCISE  32 

Prove 

1.  sin  40°  +  sin  10°  =  2  sin  25°  cos  15°. 

2.  sin  60°  +  sin  30°  =  V2  cos  15°. 

3.  cos  80°  —  cos  20°  =  —  sin  50°. 

4.  Sin33o°  +  sin3°  =  tanl8°.  6    sin  5  x  +  sin  x 

COS  00    -f-  COS  O  COS  5  X  -f-  COS  X 

5    cos  27°  +  cos  3°  _c  t  15o  7    cos  80°  +  cos  20° 

''  sin  27°  +  sin  3°  '   sin  80°  -  sin  20°      V3' 

_ 


101 


cos  .4  —  cos  B 

9.   Cp84a?  +  cos2a? 
sin  2x     sin  4  a? 


cos  A  —  cos  B  2 

11.  cos  20°  4-  cos  100°  4-  cos  140°  =  0. 

12.  sin  x  4-  sin 3  x  +  sin  5x  =  sm2 3x- 

am  as 

13.  Given  sin  A  =  |-  and  sin  B  =  J,  find  sin  (.4  4-  JB),  sin  (.4  —  J5),  cos 
(A  4-  -B),  cos  (.4— .B),  sin  2  ^4,  sin  2  B,  cos  2 .4,  cos  2  J3,  when  .4  and  B 
are  both  in  the  first  quadrant. 

14.  Find  the  numerical  value  of  sin  (60°  4-  30°).     Also  of  sin  60° 
4-  sin  30°.      Show   geometrically  why  sin  (60°  4-  30°)  does  not  equal 
sin  60°  4- sin  30°. 

Reduce  each  of  the  following  to  a  form  adapted  to  logarithmic  com- 
putation (that  is,  to  products  or  quotients): 

sin  37°  4-  sin  22°  sin  4  A  -  sin  2  A 

"  cos 38° -cos  16°'  cos  6  A 

17.  sin2  A  —  sin2  B. 

18.  Compute  the  value  of  the  expression  in  Ex.  16  when  A  =  14°. 
Also  of  that  of  Ex.  17  when  A  =  38°  and  B  =  24°. 

19.  Make  up  for  yourself  an  example  similar  to  Ex.  17. 


102  TRIGONOMETRY 

72.  Complex  Trigonometrical  Identities.  Besides  those 
already  arrived  at,  many  other  complex  relations  between  the 
trigonometrical  functions  may  be  proved.  Usually  these  re- 
lations are  proved  to  the  best  advantage  by  reducing  the  two 
expressions,  which  are  compared,  to  some  common  form,  and 
hence  inferring  their  identity  by  Ax.  1  (see  A»rt.  31). 

In  most  cases  it  is  best  to  reduce  given  functions  to  sine 
and  cosine. 

TT<     i      T>         ru    L  1  —  cos  2  A 

Ex.  1.    Prove  that  -  -  =  tan  A. 


2  sin  A  cos  A        cos  A 

2  sin2  A      _  sin  A 
2  sin  A  cos  J.     cos  ^4 

sin  A  _  sin  ^t 
cos  A     cos  J. 

Or  if  the  teacher  prefers,  the  proof  may  be  put  in  the  following  form : 

1  —  cos 2  A  _~L—  (1  —  2  sin2  A)  _       2  sin2  A      _  sin  A _  .        * 
sin  2  A  2  sin  A  cos  ^4         2  sin  A  cos  ^1     cos  A 

Ex.  2.     Prove  sin  (A  +  5)  sin  (A  -  5)  =  sin2  A  -  sin2  5. 
(sin  A  cos  5  +  cos  A  sin  5)  (sin  ^4  cos  B  —  cos  J.  sinB)  =  sin2  A  —  sin2  B. 

sin2  .4  cos2  B  —  cos2  ^4  sin2  B  =  sin2  ^1  —  sin2  B. 

sin2  ^1  (1  —  sin2  B)  —  (1  —  sin2  A)  sin2  5  =  sin2  A  —  sin2  B. 

sm*A  —  sin2  ^4  sin2  B  —  sin2  5  -f  sin2  A  sin2  5  =  sin2  A  —  sin2  B. 

sin2 .4  -  sin2  B  =  sin2  .4  -  sin2  B. 

73.    Functions  of  the  Angles  of   a  Triangle.      If  the  sum 

of  three  angles  is   180°,  the  functions  of  the  angles  have 
important  relations. 

Ex.     If  A  +  B  +  C  =  1 80°,  prove  that  sin  A  +  sin  B  +  sin  C 
=  4  cos  4  A  cos  4  B  cos  i  C. 


GONIOMETRY  103 

Hence  sin  %(A  +  B)  =  sin  (90°  -  i  0)  =  cos  £  (7. 

sin  A  4-  sin  5  +  sin  O=  sin  .4  +  sin  5  +  sin  [180°  —  (A  +  B}~\ 
=  sin  A  4-  sin  B  +  sin  (.4  4-  B) 
=  2  sin  i  (A  4-  5)  cos  1  (A  -  B) 
4-  2  sin  |-  (.4  4-  5)  cos  |  (.44-  5)  (Arts.  69,  71) 


=  4  cos  £  (7  cos  i  ^4  cos  |  - 

EXERCISE  33 

Prove  the  following  identities  : 

j.    cos  0  4-  sin  0  __  sin  2  0  +  1 
cos  0  —  sin  9        cos  2  B 

2.  2  cos  (45°  +  J-  .4)  cos  (45°  -  -*-  .4)  =  cos  A. 

3.  cos  (^L  4-  B)  cos  (.4  —  B)=  cos2  5  —  sin2  A 

4.  tan  (45°  +  x)  -  tan  (45°  —  a;)  =  2  tan  2  a;. 

5.  (  Vl  +  sin  x  —  Vl  —  sin  x)2  =  4  sin2  ^  x. 

6    cos  (a;  +  y)  +  cos  (x  —  y)  _  cos  (a;  —  y)  —  cos  (a; 
cos  a;  cos  y  sin  a;  sin  ?/ 

?    tan  (45°  +  4-  A)  +  tan  (45°  -  £^t)  =  C£C  ^ 
tan  (45°  4-  -J-  ^1)  -7  tan  (45°  - 


a 


sin  A        cos 

9.   cos^-sin^ 
cos  ^4  4-  sin  A 

10.    tan 


1  +  cos  0  4-  cos  2  0 

cot  0  —  1  _  1  —  sin  2  0 
cot  0  +  1  ~   cos  2  0 


11. 


—     x 

12.  —  —  ^f—  =  cos  x. 
1  4-  tan2  1  a?  ' 


If  ^1  +  B  4-  C  -  180°,  prove  that 

13.  cos  A  4-  cos  B  +  cos  (7=1  4-4sini.4sin-i-.Bsm|  C. 

14.  tan  A  4-  tan  J3  4-  tan  C=  tan  .4  tan  5  tan  C. 

15.  cos  (^4  +  B  +  0)  =  —  cos  2  (7. 


104  TRIGONOMETRY 

EXERCISE  34.    REVIEW 

1.  Given  cos  0  =  —  f  and  0  is  in  the  third  quadrant,  find  esc  6, 
cot  6,  sin  £  0,  tan  (180°  —  0),  sin  (—  0). 

2.  Given  tan  ^  x  =  2  (and  a?  acute),  find  sin  x. 

3.  Given  sin  2  x  =  ^  VB,  find  cot  ^  x. 

4.  Given  cos  |-  x  —  J,  find  sin  2  x  and  tan  2  x. 

5.  Given  cot  30°=  V3,  find  cos  15°,  esc  15°,  and  tan  15°. 

6.  Given  sin  A  =  f  and  A  acute,  cos    JB  =  i  and  5   acute,  find 
(a)  Bin(<4-.B);  (6)  cos  (A+B)  ;  (c)  cos  (-4-5);  (ef)  sin  25;  (e)  cos  2  B; 
(/)  tan  2  5;  (y)  cot  2  .4;  (fc)  tan  (J.  -  £)  ;  (i)  cot  (-4  +  .B)  ;  0')  cos  i  ^- 

7.  Given  cot  6  =  —  2  and  0  is  the  second  quadrant,  find  (a)  sec  0; 
(6)  tan  (180°  -  0)  ;  (c)  cot  (180°  +  0)  ;  (d)  cos  (-  0). 

8.  Find  sin,  cos,  tan,  cot,  of  : 


(a)    *  -     ;  (6)  (*-  0)  ;  («)   a  -      ;  (d)  ("  +  x)  '  where  *  = 


Prove  the  following  : 

1  —  cos  2  x  sin  x  +  sin  2  # 

9.  tan  x  =  —  —  •  12.   -  —  —  =  tan  x. 

sm  2x  1  +  cos  x  +  cos  2  # 


10.  tan^".  13. 


. 

sin  A  cos 

11  2  sin  ^1  —  sin  2  A  _  1  —  cos  A   14  sin  21°  +  sin  5°  _  tan 

~       '    '  cos  21°  +  cos  5°  " 


15  cos  9  0  +  cos  5  0  +  cos  0 
sin  9  0  +  sin  5  0  +  sin  0 


.  „        ,    o     ^  tan  a5  4-  cot  x  +  1      2  +  sin  2x 

16.    cos2  a;  tan2  a;  +  sm2x  cot  of  =  1.      19.   - 

tan  x  4-  cot  x  —  1   2  —  sin  2  a; 


17    cos  75°  +  cos  15°  ^ 


t 

sin  75°  -  sin  15°  cos  2  a;  -1 

18    sin^  +  sin5==coH(^_jB)>      a>    sin  (s  +  y)  =  cot  x  +  cot  y< 
cos  B  —  cos  J.  sin  (x  —  y)      cot  ?/  —  cot  x 

22.     COS  A  = 


cos  x  cos 
24.   cos  5  x  +  cos  3  #  =  2  cos  4  x  cos  #. 

25 


sin  2  a;  —  1      2  tanx  —  tan2x  — 


27. 


GONIOMETRY  105 

26.    sin  (45°  -f  x)  -f  sin  (45°  —  a?)  =  V2  cos  x. 

l-cot2(?-oA 
28.  _      _~  -  Z  =  - 


1  _  C0t2(  -  +  x  }  1  +  cot2(  ^  - 

V4   /  V4 

1  4-  cos  x  -f  cos  2  a  _  sin  a;  +  sin  2  # 
cos  x  sin  a; 

30.  cos  12  a-  +  cos  6  x  -+-  cos  4  a;  +  cos  2  a  =  4  cos  5  x  cos  4  x  cos  3  #. 


/  ,  ^0  .  x\         1  +  sin  x 

31.  tan[4o°-f-    =\  —         — 

^  2y       ^  1  —  sm  a; 

32.  (sin  a;  cos  y  —  cos  #  sin  y)2  -+-  (cos  «  cos  y  +  sin  a;  sin  y)2  =  1. 

33.  cos2  1  a;  (tan  1  x  —  I)2  =  1  —  sin  x. 


34.  Find  the  value  of  CSC<9  when  cot  (9  =  -      and  (9  is  in  quad- 

sec  6  +  sm  0  2 

rant  II. 

35.  Find  the  value  of  tan  e  +  cos  e  when  sin  0  =  -  1  and  (9  is  in  the 

cot  0  -f  sec  0  5 

3d  quadrant. 

36.  Simplify  cos  300°  -  cot  ^~  +  60°")  +  cot  150°  -  tan  f  -  |Y 

37.  Simplify  sin  660°  +  tan  (^f  -  600>)  +  cot  330°  +  cos  (-  30°). 

V2  / 

38.  Simplify  : 

(a  -  b)  sin  -  -  (a  +  6)  tan  225°  +  (a2  +  b2)  cot  ^  -  a  cos  f 

2  2  \ 

39.  If  tan  2  0  =  -2/,  find  tan  0  and  sin  0,  0  being  in  the  3d  quadrant. 

p          sin  (A  +  #)  _  tan  A  +  tan  j?  _  cot  B  +  cot  J. 
sin  (A  —  B)      tan  ^1  —  tan  B     cot  B  —  cot  A 

41.  If  J.  is  an  angle  in  the  second  quadrant  and  sin  A  =  f  ,  find  the 
value  of  sin  2  A  +  cos  2  A 

If  .4  +  B  +  (7=  180°,  prove  : 

42.  sin  A  +  sin  B  —  sin  (7=4  sin  1  J.  sin  1  5  cos  |-  C. 

43.  cot  i-J.  -f  cot  i  J5  +cot  i  C  =  cot  i  J.  cot  \  B  cot  1  C. 

44.  sin  2  J.  +  sin  2  J5  +  sin  2  (7  =  4  sin  A  sin  5  sin  (7. 

45.  cos  2  ^.  -f  cos  2  5  -h  cos  2  (7  =  —  (4  cos  A  cos  B  cos  (7+1). 

46.  tan  A  —  cot  .B  =  sec  A  esc  5  esc  C. 


106  TRIGONOMETRY 

In  a  right  triangle,  C  being  the  right  angle,  prove 
47.    sin2  -B  =^^>  49. 


2  2c  2          a+c 


48.     cos  sns-  so.   cos2 -.4  = 


21  A      &  +  c 


2  2     )          c  2  2c 

Using  sin  x  cos  x  =  \  sin  2  a>,  sin2  x  =  1~cos2a;,  cos2  x  =  1  +  C(°s2a;, 
transform : 

51.  sin2  ic cos2  a;  into  ^(1  —  cos  4  a?). 

52.  sin4  a?  cos2  a;  into  T^(l  —  cos  4  x)  —  i  sin2  2  a;  cos  2  x. 

53.  sin4  a;  cos4  x  into  an  expression  in  terms  of  the  cosines  of  even 
multiples  of  x. 

54.  sin8  a?  into  an  expression  of  the  same  general  kind  as  in  Ex.  53. 

55.  What  nation  first  used  the  formula  for  sin  1 A  ? 

56.  What  man  discovered  the  formula  for  sin  2  A  ? 

57.  Who     first     published     the     formulas     for     sin  (A  —  B)   and 
cos  (A  —  B)j  and  at  what  date  ? 


CHAPTER   VI 
OBLIQUE   TRIANGLES 

TRIGONOMETRIC   PROPERTIES   OF   OBLIQUE  TRIANGLES 

74.    Law  of  Sines  in  a  triangle.     In  any  triangle  the  sides 
are  to  each  other  as  the  sines  of  the  angles  opposite. 


In  Fig.  55  the  angles  A  and  B  are  both  acute. 

In  Fig.  56  the  angle  A  is  acute,  and  angle  ABC  obtuse. 

Let  (7D,  denoted  by  p,  be  the  altitude  in  each  triangle. 

In  Fig.  55,  in  the  rt.  A  J.  CT>,  p  =  6  sin  J.  ;  (Art.  41) 

in  the  rt.  A  CBD,  p  =  a  sin  B  ;  (Art.  41) 

.  *  .  -6  sin  A  =  a  sin  B:  (Ax.  1) 

In  Fig.  56,  in  the  rt.  AACD,  p  =  I  sin  A  ; 

in  the  rt.  A  BCD,  p  =  a  sin  (180°  -  Z  ABC) 
=  a  sin  /.A  BC.    (Art.  64) 

Hence  in  A  ABC  in  both  figures,  1}  sin  A  =  a  sin  B, 

or  a  :  1}  =  sin  A  :  sin  B. 

In  like  manner,      b  :  c  =  sin  B  :  sin  (7, 

and  a  :  c  =  sin  J.  :  sin  (7. 

Or,  collecting  results, 


a 
sin  A 


sn 

107 


sn 


7" 


108 


TRIGONOMETRY 


75.  Law  of  Tangents  in  a  triangle.  In  any  triangle  the 
sum  of  any  two  sides  is  to  their  difference  as  the  tangent  of 
half  the  sum  of  the  angles  opposite  the  given  sides  is  to  the 
tangent  of  half  the  difference  of  these  angles. 

In  a  triangle  ABC  (Figs.  55  and  56), 

a  :  1}  =  sin  A  :  sin  B.  (Art.  74) 

By  composition  and  division, 

a  4-  b  _  sin  A  4-  sin  B 
a  —  I}      sin  A  —  sin  B 

2  sin  \  (  A  +.B)  cos  \  (A  -  B} 


(Art.  71) 


Or, 
In  like  manner, 


and 


a  —  I} 

l^~c 

c  +  a 


tan^Qg 


It  is  also  helpful  to  have  a  geometric  proof  of  the  Law  of  Tangents. 
This  may  be  obtained  as  follows : 

In  a  given  triangle  ABC  (CB  >AC), 
produce  A  C  to  D,  making  CD=  CB  or  a. 
On  CB  mark  off  CE  =  AC  or  b. 
Draw  the  straight  line  DB. 


Also   EB  =  CB-CE-a-b. 

/-DCB,  being  an  exterior  angle  of 
A  ACE,  =  x  +  x  =  2x. 

Also  Z.DCB,  being  an  exterior  angle 
of  A  ACE,  =  A  +  5  (of  A 


Also, 


FIG.  57. 
Also  A  ADF  and  EFB  are  similar  (two  A  equal). 


OBLIQUE   TRIANGLES  109 


.-.  Z  AFD  =  Z  EFB.     .-.  AF1.  DB. 
In  &AFDandEFB,  DF  :  FB  =  a  +  b  :  a  —  b. 
In  A  AFD  and  AFB, 

tanz:tanZJZ^LB  =  —  :  — 
AF    AF 

By  Ax.  1,  a  +  b  :  a  —  b  =  tan  x  :  tan 

=  tan  %(A  +  -B)  :  tan 

76.    Law  of  Cosines  in  a  triangle. 
In  the  triangle  AB6,  Fig.  55,  by  geometry, 
a2  =  62  +  c2  -  2  c  x  AD. 

But  in  the  rt.  A  ACD,  AD  =  b  cos  A. 


If  ^4.  is  an  obtuse  angle,  Fig.  58,  by  geometry, 
a2  =  &2  +  c2  +  2  c  x  .AD. 

But  in  the  rt.  A  J.CD, 
J.D  =  &  cos  Z  CAD  =  &cos  (180°  -  A)  =  -  b  cos  A. 


Hence  in  either  case, 

2  6c  cos  J.  =  62  4-  c2  -  a\ 


or 


'A* -v 'B 


In  like  manner  it  may  be  proved 
that  FIG- 58' 


COSjB= 


77.  Formulas  derived  from  the  Cosine  Formula.  The  for- 
mula for  cos  A  in  Art.  76  has  a  numerator  which  is  primarily 
a  sum  and  difference,  hence  logarithms  cannot  be  used  in 
computing  numerical  values  from  it.  In  order  to  put  this 
formula  in  such  a  shape  that  its  value  can  be  computed  by 
the  aid  of  logarithms,  it  is  necessary  to  transform  the 
numerator  of  the  fraction  into  a  product.  This  is  done 


110  TRIGONOMETRY 

by  the  use  of  the  formula  for  the  cosine,  or  of  that  for  the 
sine  of  a  half  angle  (Art.  70).     Thus: 


2  be 


2  be  2  be 

_(b  +  c-\-a)(b+c  —a) 
2  be 

Let  2s  =  a-f-6  +  c;    then,  subtracting  2  a  from  each  member, 
2s  —  2a  =  b  +  c  —  a. 

Hence,  -8  cos^  A  =  Z  "$•  ~  2  ffi)  , 


or  cos 

In  like  manner, 


Also  from  Art.  70, 

2  sin2  1  ^1  =  1  -  cos  ^  =  1  - 


2  6c  2  ftc 

=  a«_  (b  -  c)2=  (a  +  b  -  c)(a-b  +  c) 
2  be  2  be 

=  (2  g  -  2  c)  (2  g  -  2  6)  =  4(8  -  b}  (s  -  c) 
2  be  2  be 


Hence,  sin  ±A  '- 

In  like  manner, 


Dividing  the  formula  for  sin  ^  A  by  that  for  cos 
Similarly, 


—  c)        -,  Rs  —  a}  (s  — 

1 


OBLIQUE   TRIANGLES  111 

EXERCISE  35 

1.  Prove  that  the  diameter  of  a  circle  circumscribed  about  a  triangle 
is  equal  to  any  side  of  the  triangle  divided  by  the  sine  of  the  angle 
opposite  that  side. 

2.  By  means  of  the  property  of  sines,  prove  that  the  bisector  of  an 
angle  of  a  triangle  divides  the  opposite  side  into  segments  which  are 
proportional  to  the  sides  forming  the  given  angle. 

3.  In  any  triangle  ABC,  prove  that  a  =  b  cos  C  +  c  cos  B.     State  this 
property  in  words.     Write  the  two  similar  formulas  for  b  and  c.     What 
does  the  above  formula  become  when  C  =  90°  ? 

4.  Prove  that  the  radius  of  an  inscribed  circle  of  a  triangle  is  equal 

to  °  sm  %A  S1P  ^  B  where  c  is  one  side  of  the  triangle  and  A  and  B 
cos^-C 

are  the  angles  adjacent  to  c,  and  C  is  the  angle  opposite  c. 

5.  Prove  sin  A  =  —  Vs(s  -  a)(s  -  b)(s  -  c)  if  s  =  a  +  b  +  c. 

uc  £ 


6.    Prove  cos  ^  = 


be 

7.  Find    the    form   to   which    the    formula    gL±J  =  tan  I  (A  +  B) 

a-b      tan  $(A—B) 

reduces,  and  describe  the  nature  of  the  triangle,  when  (I)  C  =  90°, 
(II)  A-B  =  90°,  and  B=C. 

8.  What  does  a2  =  b2  +  c2  —  2  be  cos  A  become  when  (I)  A  =  90°, 
(II)  A  =  0°,  (III)  A  =  180>?     What  does  the  triangle  become  in  each 
of  these  cases  ? 

9.  What   does  -  =  ^-^  become  when  A  is  a  right  angle  ?     When 

b      sin  B 

B  is  a  right  angle  ? 


SOLUTION   OF   OBLIQUE   TRIANGLES 

78.  Cases  in  the  Solution  of  Oblique  Triangles.  Four  cases 
occur  in  the  solution  of  oblique  triangles  according  as  the 
parts  given  are 

I.  One  side  and  two  angles. 

II.  Two  sides  and  the  included  angle. 

'III.  Three  sides. 

IV.  Two  sides  and  an  angle  opposite  one  of  them. 


112 


TRIGONOMETRY 


CASE  I.     ONE  SIDE  AND  Two  ANGLES  GIVEN 
79.   To  solve  Case  I  use  the  law  of  sines  (Art.  74),  thus : 

Subtract  the  sum  of  the  two  given  angles  from  180° ;  this  will 
give  the  third  angle. 

The  unknown  sides  may  then  be  found  by  the  following 
proportion : 

unknown  side :  known  side  =  sine  of  angle  opposite  the  unknown 
side  :  sine  of  angle  opposite  the  known  side. 

In  solving  oblique  triangles  by  the  use  of  logarithms  it  is  of  special 
importance  to  make  an  outline  or  skeleton  of  the  work  before  looking 
up  any  logarithms,  and  then  to  do  all  the  work  connected  with  the  use 
of  the  tables  together. 

Ex.  1.  Given  A  =  67°  21',  B  =  57°  48',  b  =  367.  Solve 
the  oblique  triangle  ABC. 

SOLUTION 


C  =  180°  -  (67°  21'  +  57°  48')  =  54°  51 f. 
Then  by  the  law  of  sines  (Art.  74),  (Check) 


a 
367 


sin  67°  21' 
sin  57°  48' 


sin  54°  51' 


367      sin  57°  48' 


sin  67°  21' 
sin  54°  21' 


Before  looking  up  any  logarithms  in  the  tables  the  pupil  should 
outline  the  work  as  follows: 


367  log 
67°  21'  log  sin 
57°  48'  colog  sin  .  .  .  . 

367  log 
54°  51'  log  sin  .  .  .  . 
57°  48'  colog  sin  .  .  .  . 

clog  .  .  .  . 
67°  21'  log  sin 
54°  51  'colog  sin  .  .  .  . 

a  =  log  .  .  .  . 

c  =  log  .... 

a  =  log  ... 

OBLIQU3  TRIANGLES 


113 


The  pupil  can  then  look  up  all  the  logarithms  at  once  and  fill  in  the 
above  tabulated  form.  (Any  logarithm  occurring  more  than  once  on 
being  taken  from  the  tables  should  be  entered  uniformly  wherever 
it  belongs.)  Proceeding  thus,  he  should  obtain 


367  log  2.56467 
67°  21'  log  sin  9.96541  -  10 
57°  48'  colog  sin  0.07253 
a  =  400.227  log  2.60231 


367  log  2.56467 
54°  51'  log  sin  9.91257  -  10 
57°  48'  colog  sin  0.07253 
c  =  354.625  log  2.54947 


(Check) 

c  log  2.54947 

67°  21' log  sin  9.96541 -10 
54°  51'  colog  sin  0.08743 
a  log  2.60231 


Ex.    2.      Solve   the    triangle 
B=  83.11°,  and  6=  7641. 


ABC,    given    A  =  18.29°, 


7641 
FIG.  60. 


C  =  180°  -  (18.29°  +  83.11°)  =  78.6°. 
Then  by  the  law  of  sines  (Art.  74), 


a         sin  18.29° 


7641      sin  83.11° 

7641  log  3.8832 
18.29°  log  sin  9.4967  -  10 
83°  11'  colog  sin  0.0032 
a  =  2416.11  log  3.3831 


sin  78.6° 


7641 


sin  83.11° 
7641  log  3.8832 
78.6°  log  sin  9.9913  -  10 
83.11°  colog  sin  0.0032 
c  =  7546  log  3.8777 


(Check) 
sin  18.29° 


c       sin  78.6° 

c  log  3.8777 

18.29°  log  sin  9.4967  -  10 
78.6°  colog  sin  0.0087 
a  log  3.3831 


114  TRIGONOMETRY 

The  accuracy  of  the  work  in  Exs.  1  and  2  might  also  have 
been  checked  by  use  of  the  formula  a2  =  62  +  c2  —  2  be  cos  A,  or 

,.         -,    A       +  s(s  —  a) 
of  cos  \A  =  \~ 

be 

In  general  in  solving  oblique  triangles  the  accuracy  of  the 
work  in  any  one  case  can  be  checked  by  applying  to  the 
results  obtained  one  of  the  rules  or  formulas  of  the  other 
cases. 

EXERCISE  36 
Find  the  remaining  parts  of  the  triangle,  given : 

1.  a  =  12.632,  .4  =  65°  35',  5  =  73°  18'. 

2.  a  =  300,  B  =  10°  18',  C=  35°  22'. 

3.  b  =  1000,  B  =  49°  18',  C  =  72°  50'. 

4.  c  =  1640.22,  (7=  18°  25',  B  =  52°  16'. 

5.  A=  66°  18'  36",  B  =  43°  43'  48",  c  =  .87654. 

6.  C=  100°  18'  42",  B  =  50°  40'  16",  c  =  114.682. 

7.  C=  22°  18'  24",  B  =  58°  12'  24",  a  =  1.26984. 

8.  A=  68°  15'  20",  B  =  43°  18'  36",  a  =  1.8263. 

9.  £  =  57°  23'  12",  ^1  =  54°  21'  18",  c=  .20814. 

10.  Given  a  =  5. 267,   ^1  =  30°,   5  =  45°,   solve   without    using  the 
tables. 

11.  Given  c  =  1000,  ^4  =  60°,  JB  =  45°,  find  a  and  b  without  using 
tables. 

12.  In  a  parallelogram  given  a  diagonal  d,  and  the  angles  m  and  n 
which  this  diagonal  makes  with  the  sides,  find  the  sides.     Find  the 
sides  when  d  =  14.632,  and  m  =  38°  18',  and  n  =  12°  32'. 


Using  four-place  tables,  find  the  unknown  parts,  having  given : 

13.  a  =  14.26,  A  =  52.16°,  B  =  71.11°. 

14.  c  =  200,  C  =  18.16°,  B  =  80.52°. 

15.  b  =  .7125,  A  =116.18°,  C  =  38.25°. 

16.  a  =  63.28,  B  =--  63.28°,  C=  36.82°. 

17.  6  =  4000,  B  =  17.28°,  (7  =  82.26°. 

18.  c  =  8,  -4  =  79.26°,  5  -  99.99°. 

19.  a  =  19.28,  B  =  42.8°,  C  =  19.53°. 


OBLIQUE   TRIANGLES  115 

20.  c  =  .2265,  B  =  71.28°,  A  =  52.85. 

21.  b  =  176.8,  C  =  9.82°,  B  =  68.22°. 

22.  a  ==  4812,  B  =  75.6°,  O  =  48.71. 

23.  5  =  14.267,  C  =  110.6°,  A  =  41.63°. 

24.  c  =  712.8,  B  =  44.18°,  A  =  79.22. 


Without  the  use  of  tables,  solve,  having  given : 

25.  a  =  100,  £  =  60°,  ^L  =  60°.  27.    a  =  500,  A  =  75°,  5  =  60°. 

26.  ^  =  120°,  B  =  30°,  c  =  200.  28.    6  =  200,  A  =  105°,  c  =  45°. 

Solve  Exs.  29-31  by  either  set  of  tables. 

29.  A  ship  S  can  be  seen  from  two  points  M  and  N  on  the  shore. 
The  distance  MN  is   700  ft.,   and   the  angles  SMN  and  SNM  are 
57°  42'  [57.7°]  and  75°  18'  [75.3°]  respectively.     Find  the  distance  of 
the  ship  from  M. 

30.  A  balloon  is  directly  over  a  straight  road,  and  between  two 
points  on  the  road  from  which  it  is  observed.     The  distance  between 
the  two  points  is  2652  yd.,  and  the  angles  of  elevation  of  the  balloon 
as  seen  from  the  two  points  are  58°  50'  [58.83°]  and  47°  24'  [47.4°] 
respectively.     Find  the  distance  of  the  balloon  from  each  of  the  given 
points,  and  also  the  height  of  the  balloon  from  the  ground. 

31.  Which  examples  in  Exercise  41   can  be  worked  by  Case  I  ? 
Work  such  of  these  examples  as  the  teacher  may  direct. 

32.  Make  up  some  practical  problem  which  can  be  solved  by  the 
method  of  Case  I  and  solve  it. 

" 

CASE  II.     Two  SIDES  AND  THE  INCLUDED  ANGLE  GIVEN 

80.   To  solve  Case  II  we  have  the  following  method  by  the 
use  of  the  law  of  tangents  (Art.  75) : 

Subtract  the  given  angle  from  180°;    divide  the  remainder 
by  2.     The  result  will  be  half  the  sum  of  the  unknown  angles. 

One  half  of  their  difference  may  then  be  found  by  the  follow- 
ing proportion: 
tan  \  the  difference  of  the  unknown  angles :  tan  \  their  sum 

=  difference  of  the  two  given  sides :  their  sum. 


116  TRIGONOMETRY 

Then  ^  sum  of  unknown  ^  -h  J  their  difference 

=  greater  unknown  Z. 

\  sum  of  unknown  ^—  \  their  difference 

=  smaller  unknown  Z. 
The  third  side  is  found  by  Case  L 

Ex.  1.    Given  a  =4527,  fc  =  3465,  C  =  66°  6'  28",  solve  the 
triangle.* 

a  +  b  =  7792. 
a  -  6  =  1062. 
.4  +  5  -  180°  -  66°  6'  28 

=  113°  53' 32". 
i-  (A  +  B)  =  56°  56'  46". 


By  the  law  of  tangents  (Art.  75), 

tan  %(A  -  B)  :  tan  |  (A  -f  B)  =  a  -  b  :  a  +  6, 
that  is,  tan  1  (A  -  B)  :  tan  56°  56'  46"  =  1062  :  7992. 

tan  1  (A     B}  =  1062  tan  56°  56' 46" 
7992 

1062  log  3.02612 
56°  56'  46"  log  tan  0.18659 
7992  log  3.91266  - 10  colog  6.09734  -  10 
±(A  -  B)  =  11°  32'  28"  log  tan  9.31005  -  10 


A  =  6S°  29'  14" 
B  =  45°  24'  18" 

The  side  c  may  now  be  found  by  Case  I. 

c          sin  66°  6'  28" 


Thus  we  have 


3465      sin  45°  24' 18" 


*  If  only  the  third  side,  c,  is  required,  and  the  numbers  representing  the  other 
sides,  a  and  &,  are  small,  the  solution  may  often  be  readily  effected  by  the  formula 
of  Art.  76  without  the  use  of  logs. 

Thus  given  a  =  5,  6  =  6,  G  =  60°,  find  c. 


c  =  Va2  +  b*  -  2  ab  cos  C  =  \/25  +  36  -  60  x  \  =  V31  =  5.5775. 


OBLIQUE   TRIANGLES  117 

3465  log  3.53970 

66°  6'  28"  log  sin  9.96109  -  10 
45°  24'  18"  log  sin  9.85254  -  10  colog  sin  0.14746 

c  =  4448.9  log  3.64825 
(What  checks  can  you  suggest  for  the  work  ?) 

Ex.2.    Given  c=  30.15,  a  =  18.159,  £=54.22°,  solve   the 
triangle. 

c  +  a  =  48.309. 

(>-  a  =  11.991. 
<7  +  .4  =  180°  -54.22° 
=  125.78°. 


By  Art.  75, 

tan  £  (<7  -  A)  :  tan  -j-  (C  +  ^4)  =  c  -  a  :  c  +  a ; 
that  is,  tan  £  (C  -  .4)  :  tan  62.89°  =  11.991  :  48.309. 

tan  ?  rc    ^n  = 11-991  tan  62>89° 

.  48.309 

11.991  log  1.0789 
62.89°  log  tan  0.2908 
48.309  log  1.6840  colog  8.3160  -  10 


$(C-A)  =  25.87°  log  tan  9.6857  -  10 

|  (C  +  A)  =62.89° 

$(C-A)  =  25.87° 

(7=88.76° 

^4  =  37.02° 

The  side  b  may  now  be  found  by  Case  I. 

b       =  sin  54.22° 
18.591  ~  sin  37.02° 

18.159  log  1.2591 
54.22°  log  sin  9.9092  -  10 
37.02°  log  sin  9.7797  -  10  colog  sin  0.2203 
b  =  24.467  log  1.3886 
(What  checks  can  you  suggest  for  the  work  ?) 


118  TRIGONOMETRY 

EXERCISE  37 

Using  five-place  tables,  solve  the  following  triangles,  having  given: 

1.  a  =  27.7,  b  =  18.6,  C  =  68°. 

2.  b  =  400,  c  =  250,  A  =  68°  18'. 

3.  A  =  30°  12'  20",  b  =  .24135,  c  =  .35627. 

4.  B  =  63°  35'  30",  a  =  .062788,  c  =  .077325. 

5.  A  =  123°  16'  30",  b  =3.1625,  c  =  3.1536. 
•      6.    A  =  52°  6',  6  =  420,  c  =  200. 

7.    (7  =  60°,  6  =  9,  a  =  7.     Find  c  only. 


SUGGESTION.       c=  Va*  -f  62— 2  a6  cos  (7. 

8.  c  =  26.369,  b  =  17.268,  ^  =  32°  18'  30". 

9.  B  =  168°  18'  39",  c  =  186.27,  a  =  132.91. 


Using  four-place  tables,  solve  the  following  triangles,  having  given : 

10.  a  =  200,  b  =  260,  C  =  51.82°. 

11.  b  =  1.763,  c  =  1.112,  A  =  28.16°. 

12.  a  =  .§782,  c  =  .412,  jB  =  112.18°. 

13.  b  =  11.65,  a  =  8.26,  (7  =  12.12°. 

14.  a  =  1720,  c  =  642,  B  =  78.63°. 

15.  b  =  9,  c  =  6,  ^4  =  60°.     Find  a  only. 


SUGGESTION.      a  =  V&2  +  c2  —  2  6c  cos  A 

16.  c  =  V7,  ft  =  VlT,  ^  =  1688°.     Find  C,  JB,  and  a. 

17.  b  =  79.23,  a  =  100.6,  C  =  68.25°. 

18.  a  =  1200,  6  =  2100,  C  =  43.18°. 


19.  a  =  12,  c  =  15,  B  =  45°.     Find  b  without  the  use  of  tables. 
Solve  the  following,  using  either  set  of  tables: 

20.  Two  trees  M  and  P  are  on  opposite  sides  of  a  pond.      The  dis- 
tance of  M  from  a  point  K  is  159.6  ft.,  the  distance  of  P  from  K  is 
216.8  ft,,  and  the  angle  MKP  is  75°  18'  [75.3°].     Find  the  distance 
between  the  trees. 


OBLIQUE   TRIANGLES  119 

21.  The  length  of  a  lake  subtends  at  a  certain  point  an  angle  of 
120°,  and  the  distances  of  this  point  from  the  two  extremities  of  the 
lake  are  2  and  3  miles  respectively.     Find  the  length  of  the  lake. 

22.  The  point  0  is  acted  on  by  a  force 
OA  of  12  pounds  and  a  force  OB  of   17 
pounds,  and  the  angle  between  the  lines 
of  direction  of  the  two  forces  is  120°  43' 
[120.72°].    What  will  be  the  resultant  force 
and  what  angle  will  it  make  with  each  of 
the  original  forces  ?      (Use  the  principle 
of  the  parallelogram  of  forces.) 

23.  Two  trains  leave  the  same  station  at  the  same  time  on  straight 
tracks  intersecting  at  an  angle  of  21°  12'  [21.2°].     If  the  trains  travel 
at  the  rate  of  40  and  50  miles  an  hour  respectively,  how  far  apart  will 
they  be  in  10  minutes  ? 

24.  The  sides  of  a  parallelogram  are  172.43  and  101.31  and  the 
angle  included  by  them  is  61°  16'  [61.27°].     Find  the  two  diagonals. 

25.  In  Exercise  41  which  examples  can  be  worked  by  the  methods 
of  Case  II  ?     Work  such  of  these  as  the  teacher  may  direct. 

26.  Make  up  some  practical  problem  which  can  be  solved  by  the 
method  of  Case  II  and  solve  it. 


CASE  III.     THREE  SIDES  GIVEN 

81.  The  Solution  of  Case  III  is  effected  by  the  use  of  the 
formulas  proved  in  Art.  77. 

In  case  it  is  desired  to  find  only  one  of  the  angles  of  a 
given  triangle  it  will  be  best  to  use  that  one  of  the  formulas 
of  Art.  77  which  will  give  t-he  required  angle  most  accurately. 
The  cosine  formula  may  be  stated  in  general  language  thus: 

The  cosine  of  one  half  of  any  angle  of  a  triangle  is  equal  to 
the  square  root  of  one  half  the  sum  of  the  three  sides  multiplied 
by  one-half  the  sum  minus  the  side  opposite,  divided  by  the 
product  of  the  other  two  sides.  Thus 


ab 


120 


TRIGONOMETRY 


Ex.  1.    If  in  the  triangle  ABC,  a=  123,  b  =  113,  c=  103, 
find  the  angle  A. 

,  =  1(123  +  113  +  103)  =  169.5. 
s  -  a  =  169.5  -  123  =  46.5. 


/169. 
"  \ 


5x46.5 


113xl03 

169.5  log  2.22917 
46.5  log  1.66745 
113  colog  7.94692-10 
103  colog  7.98716-10 
2)19.83070-20 
i  A  =  34°  37'  22"  log  cos  9.91535-10 

.-.  Z  A  =  69°  14'  44". 


In  case  the  half  angle  (^  ^4.)  to  be  computed  is  small,  it  is 
best  not  to  use  the  formula  for  cos  \  A.  Why  ? 

In  case  the  half  angle  to  be  computed  is  close  to  90°,  it  is 
best  not  to  use  the  formula  for  sin  1  A.  Why  ? 

In  case  it  is  desired  to  find  all  three  angles  of  a  triangle,  it 
is  best  to  use  the  tangent  formula  of  Art.  77.  For  it  will 
be  found  that  by  that  method  it  is  necessary  to  employ  the 
logarithms  of  but  four  different  numbers,  whereas  by  either 
of  the  other  formulas  it  is  necessary  to  use  the  logarithms  of 
seven  different  numbers.  It  is  a  further  advantage  to  trans- 
form the  tangent  formula  thus  : 


tan  i  A  = 


s(s  —  a) 


Let 


s  —  a 


.    Then 


t  an  l  A  =  —  ,  t  an  \  B  =  —  ,  t  an  l  C  = 

s—a  s—b 


s—c 


To  test  the  accuracy  of  the  work  add  the  angles  obtained. 
Their  sum  should  differ  very  slightly  from  180°. 


OBLIQUE   TRIANGLES 


121 


s  =  169.5. 
s  —  a  =  46.5. 


s  —  c  =  66.5. 


Ex.  2.     If  in  the  triangle  ABC,  a  =  123,  6  =  113,  c=103, 
find  the  three  angles  of  the  triangle. 

46.5  log  1.66745 

56.5  log  1.75205 

66.5  log  1.82282 

169.5  colog  7.77083-10 

2)3.01315 
rlog  1.50658 

r  log  1.50658 
56.5  colog  8.24795 -10 
1 5=29°  36'  25"  log  tan  9.75453-10 


.  _  J46.5  x  56.5  x  66.5 
:  ^  169.5 

r  log  1.50658 
46.5  colog  8.33255-10 


i^=34°37'22"logtan9.83913-10 

r  log  1.50658 
66.5  colog  8.17718-10 
i  (7=25°  46'  15"  log  tan  9.68376-10 


Hence  A=  69°  14' 44" 
B=  59°  12' 50" 
C=  51°  32' 30" 


180°   0'   4"     (check) 

The  fact  that  the  sum  of  the  angles  of  the  triangle  as 
computed  differs  from  180°  by  four  seconds  is  due  to  the  fact 
that  the  logarithms  used  are  only  approximately  correct  in 
the  last  figure.  When  five-place  tables  are  used,  as  in  the 
above  solution,  the  sum  of  the  angles  obtained  should  not 
differ  from  180°  by  more  than  six  or  seven  seconds. 

Ex.  3.  Find  the  three  angles  of  the  triangle  in  which 
a=  26.16,  6  =  29.15,  c=32.24. 


s  =  43.775    8-6  =  14.625 
s  -  a  =  17.615     s  -  c  =  11.535 


•.  r  = 


r.615x  14.625x11.535 
43.775 

r  log  0.9159 
17.615  colog  8.7541-10 


|  .4=25.07°  log  tan  9.6700-10 

r  log  0.9159 
14.625  colog  8.8349-10 
15=29.39°  log  tan  9.7508-10 


17.615  log  1.2459 
14.625  log  1.1651 
11.535  log  1.1620 
43.775  colog  8.3587-10 

2)1.8317 
r  log  0.9J59 

r  log  0.9159 
11.535  colog  8.9280-10 
C=35.54°  log  tan  9.8539-10 

^=50.14° 
5-58.78° 
0=71.08° 

180°     (check) 


122 


TRIGONOMETRY 


EXERCISE  38 

By  use  of  five-place  tables  solve  each  of  the  following  triangles,  hav- 
ing given: 

a  =  100, 


1. 


2. 


3. 


5. 


a  =  .117, 
\  b  =  .261, 
|  c  -  .217. 

[  a  =  122.6, 
J  b  =  169.4, 


7. 


8. 


13. 


c  =  95.2. 

a"=  79.38, 
b  =  48.16, 
c=50. 


j  b  =  125, 
( c  =  140. 

f  a  =  1.57, 
b  =  1.7, 
c  =  1.266. 

a  =  17.03, 
b  =  12.585, 
c  =  11.085. 

fa=113, 
b  =  147, 


14. 


9. 


10. 


11. 


12. 


a=  V14, 
6  =  V19, 
c  =  V33. 

fa  =  4.1409, 

6  =  4.9935, 

[  c  =  1.8181. 

(a  =  2.6, 

|  6  =5.7, 
[  c  =  7.8. 

fa  =  17.51, 
|  6  =  12.575, 
I  c  =  23.645. 


Find  the  largest  angle. 


15.  The  sides  of  a  triangle  are  10,  17,  and  25.     Find  the  smallest 
angle  in  the  triangle. 

16.  The  sides  of  a  triangle  are  3,  4,  and  5.5.     Find  the  sine  of  the 
smallest  angle. 

17.  The  sides  of  a  triangle  are  1.1, 1.3,  1.6.     Find  the  cosine  of  the 
largest  angle. 

18.  The  sides  of  a  triangle  are  18,  21,  and  25  ft.     Find  the  length 
of  the  perpendicular  from  the  vertex  of  the  largest  angle  to  the  opposite 
side. 

19.  By  use  of  four-place  tables  solve  Exs.  1-18. 

20.  The  distances  between  three  towns,  P,  Q,  R,  are  as  follows :  PQ= 
51,  QJ?=65,  P72=20.     If  R  is  due  east  from  P,  what  is  the  direction 
of  each  place  from  every  other  place?     If  R  is  N.E.  from  P,  what 
would  each  of  these  directions  be  ? 

21.  What  angle  is  subtended  by  an  island  2  miles  long  as  viewed 
from  a  point  3  miles  distant  from  one  end  of  the  island  and  4  miles 
from  the  other  end  ? 

22.  Make  up  two  practical  problems  which  can  be  solved  by  the 
method  of  Case  III  and  solve  them. 


OBLIQUE   TRIANGLES  123 

CASE  IV.     GIVEN  Two  SIDES  AND  AN  ANGLE  OPPOSITE 

ONE  OF  THEM 

82.  The  Solution  of  Case  IV,  like  that  of  Case  I,  is  effected 
by  the  use  of  the  law  of  sines  (Art.  74).  But  it  has  been 
shown  in  geometry  that  when  two  sides  and  an  angle  oppo- 
site one  of  them  are  given,  sev- 
eral special  cases  arise  in  the  con- 
struction of  the  triangle. 

Thus  in  the  triangle  ABC  (Fig. 
64)    let   the    given   parts   be  the 
angle   A    and  the  sides  a  and  b. 
Then  under  the  following  conditions  the  following  triangles 
may  be  constructed : 

I.  If  given  Z  A  is  obtuse 

and  1.    side  opp.  A  >  side  adj.     .     ...     .     .     one  A. 

2.    side  opp.  A  <  side  adj no  A. 

II.  If  given  Z.  A  is  right  (same  results  as  in  I). 

III.  If  given  Z  A  is  acute 

and  1.    side  opp.  >  side  adj one  A. 

2.  side  opp.  =  side  adj.        .     .     .  one  isosceles  A. 

3.  side  opp.  <side  adj. 

The  case  last  mentioned  (3)  subdivides  into  three  special  cases  as 
follows : 

(1)  side  opp.  >  (side  adj.)  x  (sin  given  Z.)   .     >     •        two  A. 

(2)  side  opp.  =  (side  adj.)  x  (sin  given  Z.)    .     .     one  right  A. 

(3)  .side  opp.  <  (side  adj.)  x  (sin  given  Z)    .     .     .     .   fno  A. 

In  practice,  the  cases  of  no  solution  and  of  one  right  tri- 
angle or  one  isosceles  triangle  as  the  solution  do  not  often 
occur.  Hence  we  usually  need  merely  a  method  of  discrimi- 
nating between  the  cases  where  one  oblique  triangle  or  two 


124  TRIGONOMETRY 

oblique  triangles  form  the  solution.     We  may  state  this  test 
in  the  form  of  question  and  answer  thus : 

Q.    In  general,  when  are  there  two  solutions  in  Case  IV  f 

Ans.  When  the  side  opposite  the  given  angle  is  less  than 
the  other  given  side. 

Q.  In  this  case,  how  may  the  two  triangles  be  con- 
structed f 

Ans.  Take  the  vertex  between  the  two  given  sides  as  a  center, 
and  describe  an  arc,  using  the  smaller  side  as  radius. 

'  It  is  usual  so  to  letter  the  figure 
that  the  vertex  of  the  given 
angle  comes  at  the  left  end  of 
the  unknown  base.  Thus  given 
ZC  =  38°,  b=  152,  c=  103,  we  have 
.  Fig.  65.  . 

FlG  65  Hence,  in  solving  examples  in 

Case  IV, 

Observe  whether  the  side  opposite  the  given  angle  is  less  than 
the  other  given  side;  if  it  is,  there  are,  in  general,  two  solutions, 
which  construct  by  taking  the  vertex  between  the  given  sides 
as  a  center  and  describing  an  arc  with  the  smaller  side  as 
radius. 

In  either  case  find  the  'unknown  angle  opposite  the  known 
side  by  the  use  of  the  following  proportion : 
sine  of  unknown  Z  opp.  known  side :  sine  of  known  Z 

=  side  opp.  unknown  Z :  side  opp.  known  Z. 

In  case  there  are  two  solutions,  use  in  one  triangle  the  angle 
obtained  from  the  table,  and  in  the  other  triangle  the  supplement 
of  this  angle. 

Find  the  third  angle  and  third  side  by  Case  I. 

Ex.  1.  Given  a  =  84,  fc  =  48.5,  ^  =  21°  31',  solve  the  tri- 
angle. 


OBLIQUE   TRIANGLES 


125 


Since  the  side  opposite  the  given  angle,  84,  is  greater  than  the  other 
given  side,  48.5,  there  is  but  one  solution. 

sin  B          48.5 


.'.  sin  B  = 


sin  21°  31'        84 

48.5  sin  21°  31' 


84 


48.5  log  1.68574 
21°  31 'log  sin  9.56440 -10 
84  log  1.92428  colog  8.07572  - 10 
B  =  12°  13'  33"  log  sin  9.32586  - 10. 


=  146°  15' 27". 
By  Case  I  we  find  c  =  127.211. 


Ex.  2.   a  =  22,  b  =  34,  A  =  30°  20',  solve  the  triangle. 

Since  the  side  a  opposite  the  given  angle  A  is  less  than  the  other 
given  side  (A  being  acute,  and  22  >  34  sin  30°  20')  there  are  two  solu- 
tions to  the  given  triangle.  In  this  case  it  is  well  to  draw  the  smaller 
triangle  separately  as  well  as  the  general  figure. 


C= 


C'= 


FIG.  67. 

By  the  law  of  sines  (Art*.  74), 

sin  .5      _34 
sin  30°  20' ~  22* 

34  log  1.53148 
30°  20'  log  sin  9.70332  - 10 
22  log  1.34242  colog  8.65758  -  10 


FIG.  67 a. 


sin  B  = 


34  sin  30°  20' 

22 


B  =  51°  18'  27"  log  sin  9.89238  - 10 
,'.  on  Fig.  67a,  £'=180°-51°  18' 27" 
-128°  41 '33"-. 


To  complete  the  solution  of  A  ACS, 
Z  ACE  =  180°  -(Z.A  +  ZABC) 
=  180° -81°  38' 27" 
=  98°  21'  33". 
Hence  by  Case  I  we  find 
c  =  43.098. 


To  complete  the  solution  of  A  AC'B'  (Fig.  67a). 


=  180°  -  159°  1'  33"  =  20°  58'  27". 
Then  by  Case  I  we  find  c'  =  15.5926. 
(What  checks  can  be  used  in  the  case  of  each  of  the  two  triangles  ?) 


126 


TRIGONOMETRY 


Ex.  3.     Given  a  =  22,  b  =  34,  A  =  30.33°,  solve  the  triangle. 

Since  the  side  a  opposite  the  given  angle  A  is  less  than  the  other 
given  side  (A  being  acute  and  22»>  34  sin  30.33°),  there  are  two 
solutions.  In  this  case  it  is  well  to  draw  the  smaller  triangle  separately 
as  well  as  the  general  figure. 


C'  = 


FIG.  68. 

By  the  law  of  sines  (Art.  74), 

sin  B     =  34 
sin  30.33°  "  22' 

34  log  1.5315 
30.33°  log  sin  9.7033  -  10 
22  log  1.3424  colog  8.6576  -  10 
B  =  51.32°  log  sin  9.8924  -  10 


sm  ±f  = 


34  sin  30.33C 

22 


To  complete  the  solution  of  AACB, 
^ACB  =  180°  -  (30.33°  +  51.32°) 

=  98.35°. 
Hence  by  Case  I,  obtain  c  =  43.1. 


=  180°  -  51.32°  =  128.68°. 
To  complete  the  solution  of  AAC'B'  (Fig.  68a), 
we  have  0"  =  180°  -  (30.33°  +  128.68°)  =  20.99°. 

Hence,  by  Case  I,  find  c'  — 15.6. 

EXERCISE  39 

State  the  number  of  solutions  for  each  of  the  following  and  con- 
struct a  figure  for  each  example,  lettering  it  according  to  the  method 
specified  in  Art.  82 : 

5.  (7=80°,  6  =  16,  c  =  15.5. 

6.  5  =  54°,  a  =  23,  6  =  36. 

7.  (7  =  30°,  a  =  18,  c  =  9. 


1.  .4  =  30°,  6  =  50,  a  =  60. 

2.  .6  =  30°,  a  =  100,  6  =  70. 

3.  (7  =  45°,  a  =  60,  c  =  60. 

4.  A  =  60°,  6  =  12,  a  =  10. 


8.   5  =  50°,  a  =  50,  6  =  37. 


9.   J.  =  75.16°,  c  =  18,  a  =  17.6. 

Using  five-place  tables,  solve  the  following  triangles,  having  given : 

10.  A  =  38°  18',  6  =  120.6,  a  =  138.7. 

11.  .4  =  61018';  c  =  23.7,  a  =  21.25. 


OBLIQUE   TRIANGLES  12? 

12.  (7  =  104°  13'  48",  6  =  115.72,  c  =  165.28. 

13.  B  =  22°  22',  a  =  .6728,  6  =  .81434. 

14.  ^1  =  47°  19',  a  =  100,  c«=120. 

15.  B  =  15°  30'  12",  a  =  1200,  6  =  590. 

16.  C  =  78°  18'  18",  a  =.26725,  c  =  . 37926. 

17.  £  =  26°  18'  36",  a  =  28.604,  6  =  12.678. 

18.  A  =  131°  18'  24",  a  =  .8888,  c  =  .4128. 

19.  (7  =  31°  31' 15",  6  =  11.111,  c  =  8.267. 


Using  four-place  tables,  solve  the  following  triangles,  having  given : 

20.  B  =  32.37°,  6  =  126.6,  a  =  138.7. 

21.  xL  =  57.366°,  c  =  22.7,  a  =  20.672. 

22.  £  =  105.273°,  6  =  306.72,  c  =  241.8. 

23.  (7  =  26.223°,  a  =  66.35,  c  =  82.59. 

24.  5  =  14.3°,  a  =  20.17,  6  =  17.8. 

25.  .4  =  22.37°,  c  =  300,  a  =  200. 

26.  5  =  63.31°,  c  =  7.67,  6  =  9.54. 

27.  (7  =  49.31°,  6  =  .17634,  c  =  . 15678. 


28.  In  a  parallelogram,  one  side  is  167,  one  diagonal  is  295.6,  and  the 
angle  included  by  the  diagonals  is  24°  18'  [24.3°].     Find  the  other  side 
and  other  diagonal,  and  also  the  angles  of  the  parallelogram. 

29.  If  the  angle  between  two  forces  is  154°  20'  [154.33°],  one  of  the 
forces  is  960  pounds,  and  the  resultant  of  the  two  forces  is  440.46 
pounds,  find  the  other  force. 

AREA  OP  AN  OBLIQUE  TRIANGLE 

83.  I.  Given  two  sides  and  the  included  angle,  to  find  the 
area  of  a  triangle,  use  the  rule : 

The  area  of  a  triangle  equals  one  half  the  product  of  any  two 
sides  multiplied  by  the  sine  of  the  angle  included  by  these  sides. 

For  let  the  given  sides  be  a  and  c. 


128 


TRIGONOMETRY 


In  Fig.  69a,  let  /.B  be  acute  ;  in  Fig.  696,  let  Z. ABC  be 

obtuse. 

c  c 


Let  p  be  the  perpendicular  from  C  to  A B  or  AB  produced. 
In  each  figure,  the  area  of  A  ABC  =  \c  x  p. 
In  Fig.  69a,  in  the  rt.  ACBD,  p  =  a  sin  B.  (Art.  41) 

In  Fig.  696  in  the  rt.  A  CBD,  p  =  asm  (180°-Z^£C) 

=  a  sin  ABC.         (Art  64) 
Hence,  in  each  figure,  if  we  denote  area  of  A  ABC  by  K, 

K=^ac  sinB. 

In  case  the  given  parts  are  a,  b,  C9  or  6,  c,  A,  let  the  pupil 
state  what  the  formula  becomes. 

Let  the  pupil  also  state  these  formulas  in  general  language. 

Ex.  1.     J.  =  66°  4'  19",  6  =21.66,  c=  36.94,  find  the  area 
of  the  triangle  ABC. 

By  the  formula  K  =  \  be  sin  A, 

K=  1(21.66  x  36.94  x  sin  66°  4'  19"). 
.-.  log  K=  log  21.66  +  log  36.94  +  log  sin  66°  4'  19" 

+  colog  2. 

21.66  log  1.33566 

36.94  log  1.56750 
66°  4'  19"  log  sin  9.96097  -  10 

2  colog  9.69897  -  10 

Area=  365.682  log  2.56310 


21.66 
FIG.  70. 


Ex.  2.     Given  A  =  66.07°,  b  =  21.66,  c  =  36.94,  find  the  area 
of  the  triangle  ABC. 


OBLIQUE   TRIANGLES  129 

By  the  above  rule, 

K=  i  (21.66  x  36.94  x  sin  66.07°). 

.-.  log  K=  log  21.66  +  log  36.94  +  log  sin  66.07°  +  colog  2. 
21.66  log  1.3357 
36.94  log  1.5675 
•  66.07°  log  sin  9.9610  -  10 
2  colog  9.6990  —  10 
Area  =  365.75  log  2.5632 

84.    IT.    Given  two  angles  and  a  side,  find   the  third  angle 
as  usual.     Let  the  given  side  be  a,  then  a.  second  side  c  may 
be  determined  as  follows  : 
c  :  a  =  sin  C  :  sin  A. 

_  a  sin  C  _  a  sin  C  _  _     a  sin  C 

'  sin  A  ~  sin  [180°  -(B  +  C)]  "  sin  (B  +  C) 

Substituting  this  result  in  the  formula  for  K  in  Art.  83, 
ft*  sin  B  sin  C 


Hence  the  area  may  be  found  by  substituting  directly  in 
this  last  formula. 

85.    III.    Given  three  sides.     In  this  case  we  know  from 
plane  geometry  that 


-  a)(s  —  b)(s  -  c). 

86.  IV.  In  case  two  sides  and  an  angle  opposite  one  of  them 
are  given,  to  find  the  area  it  is  necessary  to  find  the  log  sin  oi 
the  angle  included  between  the  two  given  sides  by  the 
method  of  Case  IY  (Art.  82),  and  then  proceed  as  in  Art.  83, 
In  some  cases  two  answers  may  occur  (see  Art.  82). 

EXERCISE  40 

Using  either  five-place  or  four-place  tables,  find  the  area  of  the 
following  triangles,  having  given: 

1.  a  =  16.7,  6  =  21.6,  C  =  36°  18'  24"  [36.61°]. 

2.  a  =  .86,  B  =  52°  18'  [52.3°],  C  =  66°  42'  [66.7°]. 


130  TRIGONOMETRY 

3.  a  =  18,  6  =  14,  c  =  24. 

4.  6  =  200,  c  =  150,  A  =  72°  18'  30"  [72.31°]. 

5.  b  =  600,  A  =  18°  26'  [18.43°],  C=  31°  44'  [31.73°]. 

6.  b  =  14.7,  a  =  18.6,  A  =  74°  18'  [74.3°]. 

7.  a  =  .8167,  &  =  .68256,  c  =  .72623. 

8.  a  =  100,  c  =  125,  B  =  170°  16'  [1 70.27°]. 

9.  6  =  62.8,  c  =  47.2,  ^L  =  60°. 

10.  Given  ^L  =  29°  32'  16"   [29.54°],  6  =  500,  and  a=300,  find  the 
difference  in  area  between  the  two  triangles  which  contain  these  parts. 

11.  In  a  parallelogram,  given  two  adjacent  sides,  c  and  d,  and  the 
included  angle  A,  obtain  a  formula  for  the  area  of  the  parallelogram 
in  terms  of  the  given  parts. 

12.  Prove  that  the  area  of  any  quadrilateral  is  equal  to  one  half  the 
product  of  its  diagonals  and  the  sine  of  their  included  angle. 

13.  Two  sides  of  a  parallelogram  are  30  and  40  respectively,  and 
their  included  angle  is  60°.     Find  the  area  of  the  parallelogram  without 
the  use  of  tables. 

14.  The  diagonals  of  a  quadrilateral  are  17.6  and  20.5,  intersecting  at 
an  angle  of  36°  18'  [36.3°].     Find  the  area  of  the  quadrilateral. 


CHAPTER   VII 
PRACTICAL   APPLICATIONS 

87.  Instruments  for  Measuring  Angles.     In  order  to  deter- 
mine unknown  heights  or  distances  it  is  important  to  have 
an  instrument  for  measuring  angles  either  in  the  horizontal 
or  in  the  vertical  plane.     Horizontal  angles  can  be  measured 
by  the  Surveyor's  Compass.     Both  horizontal   and   vertical 
angles  can  be  measured  by  the  Transit  Instrument. 

88.  An  angle  of  elevation  is  the  angle  between  a  line  drawn 
from  the  eye  of  the  observer  to   the   point    observed   and 
the  horizontal  plane  through  the  eye  of  the  observer,  when 
this  angle  is  above  the  horizontal  plane. 

Thus,  on  Fig.  71,  ACB  is  the  angle  of  elevation  of  A  as 
viewed  from  C. 

An  angle  of  depression  is  the  angle  between  a  line  drawn 
from  the  eye  of  the  observer  to  the  point  observed  and  the 
horizontal  plane  through  the  eye  of  the  observer,  when  this 
angle  is  below  the  horizontal  plane. 

Thus,  on  Fig.  71,  DAC  is  the  angle  of  depression  of  C  as 
viewed  from  A. 

89.  I.    To  determine  the  Height  of   an  Accessible  Object 
above  a  Horizontal  Plane. 

In  Fig.  71   let  AB  be  the  object 
whose  altitude  is  sought,  and  EF  the  P. 

horizontal  plane,  and  C  the  point  of 
observation. 

In  the  right  triangle  ABC,  what 
line  shall  we  measure  ?    What  angle  ? 


How  then  can  AB  be  computed  ?  yIG.  u, 

131 


132 


TRIGONOMETRY 


D 


p> 


F 


90.  II.   To  find  the  Distance  on  a  Horizontal  Plane  to  an  In- 
accessible Object  whose  Height  is  Known.     In  Fig.  71,  let  AB 

be  the  inaccessible  object  whose  height  is  known ;  let  EF  be 
the  horizontal  plane  and  C  the  position  of  the  observer.  In 
the  right  triangle  ABC,  what  side  is  known?  What  angle 
can  be  measured  ?  How  then  can  BC  be  computed  ? 

91.  III.    To  determine  the  Height  of  an  Inaccessible  Object 
above  a  Horizontal  Plane. 

Let  AB,  Fig.  72,  be  the 
altitude  which  is  to  be  meas- 
ured, and  EF  the  horizontal 
plane.  Place  the  transit  in- 
strument at  D  and  measure 
*IG-  72'  the  angle  of  elevation  ADB. 

Measure  the  distance  DC  toward  B,  and  measure  the  angle 
ACB.  By  solving  the  triangle  ACD  the  line  AC  is  found. 
By  solving  the  right  triangle  ACB,  AB  is  found. 

In  case  it  is  desired  to  compute  AB  by  means  of  right  tri- 
angles alone,  the  solution  may  be  effected  by  dropping  a  per- 
pendicular CP  from  C  to  AD  and  solving  the  right  triangles 
DCP,  CPA,  and  CAB  (let  the  pupil  supply  the  exact  steps 
in  this  process). 

Or  we  may  proceed  by  the  use  of  natural  tangents  thus : 
On  Fig.  72,  in  A  DAB,  DB  =  AB  tan  Z  DAB, 
in  A  CAB,   CB  =  ABttmZ.CAB. 


or 


Subtracting,  DB-CB, 

DC  =  AB  (tan  Z  DAB  -  tan  Z  CAB). 
DC 


Hence 


tan  Z  D AB  -  tan  Z  CAB ' 


In  case  it  is  not  possible  to  move  directly  from  D  toward 
B,  we  may  proceed  as  follows:    Measure  Z  ADB  (Fig.  73). 


PRACTICAL   APPLICATIONS 


133 


Measure  the  line  DC  in  the 
horizontal  plane  in  any  con- 
venient direction  from  D. 
Measure  £  BDC  and  DCB. 

Then  in  the  triangle  DCB, 
DB  may  be  computed  (How?). 
Afterward  in  the  triangle 
ADB  compute  AB  (How?). 


FIG.  73. 


92.  IV.    To  determine  the  Height  of  an  Inaccessible  Object 
on  an  Inclined  Plane. 

Let  DF  (Fig.  74)  be  the  horizontal  plane,  DB  the  inclined 

plane,  and  AB  the  object  whose 
height  is  sought.  If  we  measure 
the  A  ADC  and  ACB,  and  the  dis- 
tance DC,  we  may  then  compute 
AC  (How  ?).  If  we  then  measure 
/_  BDF,  we  may  compute  /_  CAB 
(How?).  Then  AB  may  be  com- 
FIG.  74.  puted  (How?). 

93.  V.    To  find  the  Distance  of  an  Inaccessible  Object. 


Let  A  (Fig.  75)  be  the  position  of 
the  observer  and  let  it  be  required  to 
determine  the  distance  from  A  to  B. 

Let  the  pupil  determine  what  meas- 
urements and  computations  are  neces- 
sary in  accordance  with  the  figure. 


FIG.  75. 


94.  VI.   To  find  the  Distance  between  two  Objects  separated 
by  an  Impassable  Barrier  (and  possibly  invisible  to  each  other). 


134 


TRIGONOMETRY 


Let  it  be  required  to  find  the  dis- 
tance between  A  and  B  (Fig.  76), 
which  are  separated  by  a  swamp  or  a 
mountain  for  instance.  Take  a  sta- 
tion C  from  which  both  A.  and  B  are 
visible.  Measure  the  angle  C  and  the 
lines  CA  and  CB.  In  the  triangle 
ABC,  compute  AB  (How  ?). 

95.  VII.    To  find  the  Distance  between  two  Objects,  both 
Inaccessible  and  lying  in  the  Horizontal  Plane. 

Let  A  and  B  (Fig.  77)  be 
two  inaccessible  objects  (as 
two  islands  off  the  shore  CD). 
Measure  the  line  CD  and  the 
£  ACD,  BCD,  ADC,  BDC. 
In  the  triangle  ACD,  com- 
pute 'AC;  in  the  triangle 
BCD,  compute  BC ;  in  the  FIG.  77. 

triangle  ABC,  compute  AB. 

96.  Range  Finders.     In  war,  both  on  land  and  sea,  the  use 
of  a  range  finder  to  determine  the  distance  of  an  enemy  is 
becoming  general.     The  essential  principle  of  such  'an  instru- 
ment is  the  finding  of  the  distance  of  an  inaccessible  object 
by  the  solution  of  a  triangle  in  which  a  side  (called  a  base 
line)  and  the  two  angles  which  include  the  side  are  known 
(see  Art.  93).     On  land  a  convenient  base  line  is  taken  and 
measured.      In   naval   warfare,   the   distance   between    two 
points  on  the  vessel  is  utilized  as  a  base  line.     In  the  range 
finder  the  triangle  employed  is  not  usually  solved  by  numer- 
ical  computation,  but   by  some  mechanical  method,  which 
gives  the  result  sought  much  more  expeditiously. 

97.  Coast  and  Geodetic  Survey.     The  essential   parts   of 
the  work  of  the  coast  and  geodetic  survey  are  as  follows : 


PRACTICAL  APPLICATIONS 


135 


1.  The  measurement  of  a  base  line  AB  (Fig.  78)  at  least 
4  or  5  miles  long,  so  accurately  that  the  error  shall  not  ex- 
ceed -^  of  an  inch  per  mile. 

2.  The  choice  of  a  convenient  station 
P    and   the   measurement    of  the  angles 
PAB  and  PBA,  and  the  computation  of 
PA  and  PB  in  the  triangle  PAB. 

3.  The  choice  of  another  station  Q,  the 
measurement  of  the  angles  QBP  and  QPB, 
and  hence   the  computation  of   PQ  and 
QB. 

4.  Proceeding    in   like   manner   from 
station  to  station  till  convenient  points,  C 
and  D,  are  reached,  and  the  length  of  the 
line  CD  computed. 

5.  The    careful    measurement  of   CD 
and  the  comparison  of  its  computed  length 
with    the    result    of    the    measurement. 
This  final  measurement  of  CD  serves  as 
a  test  of  the   accuracy  of  all  the  inter- 
vening work.     By  carrying  these  measurements  far  enough, 
a  considerable  arc  of  a  great  circle   of  the   earth   may  be 
measured,  and  from  this  arc  the  radius  or  diameter  of  the 
earth  computed. 

98.  Distance  of  the  Sun  and  Stars.  The  usual  method  of 
determining  the  distance  of  the  sun  from  the  earth  consists 
essentially  in  taking  a  line  (AB,  Fig.  79)  nearly  equal  to  the 

diameter  of  the  earth  as  a  base 
line,  and  observing  from  each  end 
of  AB  the  angle  made  by  a  line 
drawn  to  some  convenient  planet 
P.  The  distance  of  the  planet 
may  then  be  computed  by  Art.  93.  The  ratio  of  the  dis- 
tance of  the  sun  to  that  of  the  planet  from  the  earth  being 


Fia.  78. 


FIG.  79. 


136  TRIGONOMETRY 

known  by  an  astronomical  law,  the  distance  of  the  sun  is 
readily  determined.  The  distance  of  tLe  sun  from  the  earth 
is  thus  found  to  be  approximately  93,800,000  miles. 

The  distances  of  the  fixed  stars  are  found  by  taking 
the  diameter  of  the  earth's  orbit  as  a  base  line,  measuring 
the  angles  made  by  this  line  with  lines  drawn  from  its  ends 
to  a  fixed  star,  and  making  the  necessary  computations. 

Thus  the  trigonometrical  solution  of  a  triangle  in  which 
a  side  and  the  two  angles  adjacent  to  it  are  known  is  seen 
to  have  very  wide  practical  applications. 

99.    Application    to    Navigation.     Trigonometry    also    has 
many    applications    to    different    departments   of    applied 
science.      As  an  illustration  of   these 
applications  we  will  briefly  indicate  its 
method  of  use  in  navigation. 

If  a  ship  should  sail  from  R  to  B  on 
the    diagram  (Fig   80),  crossing   each 
meridian  at  the  same  angle,  for  certain 
purposes  the  A  ARE  (AB  being  the  arc 
~fr  of  a  parallel  of  latitude)  could  be  re- 

FIG.  so.  garded  as  a  plane  triangle  and  solved, 

when  necessary,  by  the  methods  of  plane  trigonometry. 
This  form  of  navigation  is  called  Plane  Sailing. 

The  departure  between  two  meridians  is  the  arc  of  a  par- 
allel of  latitude  comprehended  between  the  two  meridians. 
Thus,  AB  is  a  departure  between  PAP'  and  PBP'.  Evi- 
dently the  departure  between  two  given  meridians  diminishes 
with  the  distance  from  the  equator. 

The  difference  of  longitude  between  two  places  is  the  angle 
at  the  pole  (or  the  arc  on  the  equator)  included  between  the 
meridians  of  the  two  given  places.  Thus  the  difference  of 
longitude  for  A  and  D  is  the  angle  RPS,  or  arc  RS. 

In  Parallel  Sailing  a  vessel  sails  due  east  or  west  (i.e. 
on  a  parallel  of  latitude)  as  from  A  to  B.  The  difference  of 


PRACTICAL   APPLICATIONS  137 

longitude  corresponding  to  the  course  sailed  may  be  found 
by  the  formula 

diff.  of  longitude  —  departure*  sec-  latitude. 

For  on  Fig.  80, 

n  4 
diff.  long.  :  dep.  -  arc  RS  :  arc  AB  =  OR  :  CA  =  OA  :  CA  =  ^j  :  1 

C-OL 

=  sec.  lat  :  1. 
.-.  diff.  long.  :  departure  =  sec.  lat.  :  1. 

In  Middle  Latitude  Sailing  a  ship  sails  between  two  places 
in  a  course  oblique  to  a  parallel  of  latitude.  For  short  dis- 
tances (especially  near  the  equator)  sufficient  accuracy  is 
obtained  by  regarding  the  departure  as  measured  on  the 
parallel  of  latitude  midway  between  the  parallels  of  the  two 
places,  and  computing  the  difference  of  longitude  by  the 

formula  .  =  departure  x  sec.  mid.  lat. 


EXERCISE  41 

1.  In  Exercise  22  point  out  the  examples  which  are  solved  by  the 
method  of  Art.  89. 

2.  Also  those  which  are  solved  by  the  method  of  Art.  90. 

3.  Also  those  solved  by  principles  contained  or  implied  in  Art.  91. 

4.  The  angle  of  elevation  of  the  top  of  a  tree  measured  from  a 
point  213.5  ft.  from  its  foot  is  observed  to  be  18°.     Find  the  height, 
of  the  tree. 

5.  A  water  tower  92.5  ft.  high  stands  on  a  horizontal  plane.     An 
observer  finds  the  angle  of  elevation  of  the  top  of  the  tower  to  be  52°. 
Find  the  distance  of  the  observer  from  the  base  of  the  tower. 

6.  Pike's  Peak  when  viewed  from  a  certain  point  on  the  Colorado 
plain  has  an  angle  of  elevation  of  15°  48'  [15.8°].     Two  miles  farther  off 
the  angle  of  elevation  is  11°  59'  [11.98°].     What  is  the  altitude  of  the 
mountain  above  the  Colorado  plain  ?     If  the  Colorado  plain  is  5176  ft. 
above  sea  level,  what  is  the  altitude  of  Pike's  Peak  above  sea  level  ? 

7.  From  the  top  of  a  hill  350  ft.  high  the  angle  of  depression  of 
the  top  of  a  tower  which  is  known  to  be  150  ft.  high  is  57°.     What  is 
the  distance  from  the  foot  of  the  tower  to  the  top  of  the  hill  ? 


138  TRIGONOMETRY 

8.  A  man  standing  west  of  a  tree,  on  the  same  horizontal  plane, 
observes  its  angle  of  elevation  to  be  48° ;  he  goes  north  50  yd.  and  finds 
its  angle  of  elevation  to  be  41°.     Find  the  height  of  the  tree. 

9.  The  angle  subtended  by  a  tower  on  an  inclined  plane,  is  at  a 
certain  point  on  the  plane  56° ;  200  ft.  further  down  it  is  28°.     The 
inclination  of  the  plane  is  7°.     Find  the  height  of  the  tower. 

10.  From  the  top  and  bottom  of  a  castle  which  is  75  ft.  high  the 
angles  of   depression  of  a  ship  at  sea  are  19°  and  15°  respectively. 
Find  the  distance  of  the  ship  from  the  bottom  of  the  castle. 

11.  A  monument  70  ft.  high  and  a  tower  stand  on  the  same  hori- 
zontal plane.     The  angle  of  elevation  of  the  top  of  the  tower  at  the  top 
of  the  monument  is  20°  40'  12"  [20.67°],  at  the  base  of  the  monument 
it  is  53°  31'  12"  [53.52°].     Find  the  height  of  the  tower  and  its  dis- 
tance from  the  monument. 

12.  The  three  angles  of  a  triangle  are  to  each  other  as  11  :  13  :  6 
and  the  longest  side  is  11.     Find  the  other  two  sides. 

13.  Two  mountains,  A  and  B,  are  respectively  12  and  16  mi.  from 
a  point  O,  and  the  angle  ACB  is  72°  18'  [72.3°].     Find  the  distance 
betweeh  the  mountains. 

14.  In  a  parallelogram  one  side  is  16.9  and  a  diagonal  is  30.72,  and 
the  angle  included  by  the  diagonals  is  26°  36'  [26.6°].     Find  the  other 
side  and  the  other  diagonal,  also  the  angles  of  the  parallelogram. 

15.  A  flagstaff  50  ft.  in  height  stands  on  a  tower.     From  a  position 
near  the  base  of  the  tower,  and  on  the  same  horizontal  plane,  the  angles 
of  elevation  of  the  top  and  bottom  of  the  flagstaff  are  41°  36'  [41.6°] 
and  22°  18'  [22.3°],  respectively.     Find  the  distance  and  height  of  the 
tower. 

16.  The  diagonals  of  a  parallelogram  are  12.5  and  12.8  ft.  respec- 
tively, and  their  included  angle  is  52°  16'  [52.27°].     Find  the  sides  of 
the  parallelogram. 

17.  The  sides  of  a  triangle  are  11,  13,  and  16.     Find  the  cosine  of 
the  largest  angle. 

18.  From  a  point  4  mi.  from  one  end  of  an  island  and  7  mi.  from  the 
other,  the  island  subtends  an  angle  of  33°  33'  33"  [33.56°].     Find  the 
length  of  the  island. 

19.  Two  buoys  are  1500  yd.  apart.     The  angles  formed  by  lines 
from  a  boat  to  each  buoy  form  angles  with  the  line  between  the  buoys 
of  77°  18'  [77.3°]  and  51°  16'  [51.27°],  respectively.     Find  the  distance 
of  the  boat  from  the  nearer  buoy. 


PRACTICAL   APPLICATIONS  139 

20.  Two  straight  roads  cross  each  other  at  an  angle  of  48°  24'  [48.4°] 
at  the  point  M.     Four  miles  from  M  on  one  road  is  the  town  of  P,  and 
6  miles  from  M  on  the  other  road  is  the  town  of  If.     How  far  apart  are 
P  and  If?     (Two  answers.) 

21.  The  diagonals  of  a  quadrilateral  are  47.6  and  61.23  rd.,  respec- 
tively, and  the  angle  included  by  the  diagonals  is  43°  10'  [43.17°]. 
Find  the  area  of  the  quadrilateral. 

22.  To  find  the  distance  between  two  trees  Tand  T',  on  opposite  sides 
of  a  river,  a  line  TK  and  the  angles  T'TIf  and  T'KT  are  measured 
and  found  to  be  412  ft.,  62°  30'  [62.5°],  and  57°  32'  [57.53°],  respectively. 
Find  the  distance  TT. 

23.  Two  objects  which  are  invisible  from  each  other  on  account  of 
a  hill  are  visible  from  a  station  whose  distances  from  the  objects  are 
367  yd.  and  514  yd.,  respectively,  and  the  angle  at  the  station  subtended 
by  the  distance  between  the  objects  is  57°  36f  [57.6°].     Find  the  distance 
between  the  objects. 

24.  Given  a  circle  with    radius   19.8  ft.     Find  the  area   inclosed 
between  two  parallel  chords  on  opposite  sides  of  the  center  whose 
lengths  are  25.6  and  31.7. 

25.  Wishing   to  find   the  distance   between   two  trees    T  and    T7', 
separated  by  a  marsh,  I  take  TK  on  the  prolongation  of  TT'  through 
T,  89  yd.  in  length,  and  then  take  KP,  165  yd.  in  length,  at  right 
angles  to  IfT.     The  angle  T'PT  is  found  to  be  33°  36'  36"  [33.61°]. 
Find  the  distance  from   T  to  T". 

26.  Two  yachts  start  at  the  same  time  from  the  same  point,  and  sail 
one  due  west  at  the  rate  of  9.75  mi.  per  hour,  and  the  other  due  north- 
west at  the  rate  of  11.5  mi.  per  hour.     How  far  apart  will  they  be  at 
the  end  of  2  hr.  sail  ? 

27.  In  order  to  find  the  distance  from  a  rock  R  to  a  buoy  B,  dis- 
tances EK  and  KP  are  measured  to  points  If  and  P  from  which  both 
rock  and  buoy  can  be  seen,  the  distance  RK  being  2500  m.,  and  KP  being 
3600  m.     The  following  angles  are  then  measured:   Z.BKR  =  38°  48' 
[38.8°],  Z£/rP  =  75°54'  [75.9°],  and  ^BPIf=79° 30'  [79.5°].     Find 
the  distance  from  the  rock  to  the  buoy. 

28.  A  ship  sails  due  east  416  mi.   in  latitude  40°  23'.     Find  the 
difference  in  longitude  which  she  makes. 

29.  A  ship  leaves  latitude  30°  16'  K,  longitude  43°  17'  W.,  and  sails 
N.E.  350  mi.     Find  the  difference  of  latitude  and  departure  which  she 
makes. 

Hence  find  her  new  latitude  and  longitude. 


140  TRIGONOMETRY 

30.  A  flagstaff  30  ft.  high  stands  on  the  top  of  a  building.     From 
a  point  on  the  ground,  the  angles  of  elevation  of  the  top  and  bottom  of 
the  flagstaff  are  observed  to  be  41°  and  36°  respectively.     Assuming 
the  ground  to  be  level,  find  the  height  of  the  building. 

31.  A  tower  stands  on  a  hillside  whose  inclination  to  the  horizon 
is  11° ;  a  line  is  measured  straight  up  the  hill  from  the  base  of  the 
tower  110  ft.  in  length  and,  at  the  upper  extremity  o.f  the  line,  the 
tower  subtends  an  angle  of  52°.     Find  the  height  of  the  tower. 

32.  A  rock  60  ft.  high  stands  on  the  top  of  a  hill  whose  side  is 
inclined  21°  to  the  horizon.     An  observer  standing  on  the  hillside  below 
the  rock  finds  the  angle  of  elevation  of  the  top  of  the  rock  to  be  64°,  and 
a  second  observer,  farther  down  the  slope,  and  in  direct  line  with  the 
first  observer,  finds  the  angle  of  elevation  of  the  top  of  the  rock  to 
be  42°.     Find  the  distance  between  the  observers,  and  the  distance 
from  the  first  observer  to  the  base  of  the  rock. 

33.  A  point  at  0  is  acted  on 
by  a  force  which  gives  a  velocity 
of  1376  ft.  per  second  along  OA, 
and  by  another  force  which  gives 
0  a  velocity  of  1135  ft.  per 
second  along  OB.  Z  AOX=  30°, 
Z  BOX  =  101°.  What  will  be 
the  magnitude  and  direction  of 
the  resultant  velocity  ? 

34.  Show  that  the  projection  of  OA  plus  the  projection  of  OB  on 
X' OX  equals  the  projection  of  the  resultant  of  OA  and  OB  on  X'OX. 

35.  If,  in  the  figure  of  Ex.  33,  OA  =  200  and  the  resultant  =  300,  find 
OB,  the  angles  being  unchanged. 

36.  A  tower  190  ft.  high  stands  on  the    seashore.     From  its   top 
the  angle  of  depression  of  two  boats  are  8°  and  11°  respectively.     From 
the  bottom  of  the  tower  the  angle  subtended  by  the  distance  between 
the  boats  is  101°.     Find  the  distance  between  the  boats. 

37.  A  man  on  the  opposite  side  of  a  river  from  two  trees  P  and  Q 
wishes  to  determine  the  distance  between  the  trees.     H,e  measures  a 
distance  A  B,  287  ft.     He  also  measures  the  angles  PAB,  QAB,  PBA, 
and  PBQ  and  finds  them  31°,  36°,  51°,  and  42°,  respectively.     Find  the 
distance  between  the  trees. 

38.  Two  straight  paths  cross  each  other  at  an  angle  of  68°.     A  line 
is  drawn  so  as  to  inclose,  with  the  two  paths,  an  acre  of  ground.     This 
line  cuts  one  of  the  paths  at  a  distance  of  52  yd.  from  the  point  of 


PRACTICAL   APPLICATIONS  141 

intersection  of  the  two  paths.     What  angle  does  this  line  make  with 
each  path  ? 

39.  A  tower   135   ft.    high   stands   at   one  corner  of   a  triangular 
garden.     From  the  top  of  the  tower  the  angles  of  depression  of  the 
other  two  corners  of  the  garden  are  56°  18'  [56.3°]  and  19°  36' [19.6°], 
respectively.     The  side  of  the  garden  opposite  the  tower  subtends,  from 
the  top  o£  the  tower,  an  angle  of  66°.     Find  the  length  of  the  sides  of 
the  garden. 

40.  Two  towers  are  144  ft.  apart.     The  angle  of  elevation  of  one 
observed  from  the  base  of  the  other  is  twice,  that  of  the  first  observed 
from  the  base  of  the  second;  but  from  a  point  midway  between  the 
towers,  the  angles  of  elevation  of  the  tops  of  the  towers  are  complemen- 
tary.    Find  the  height  of  the  towers.     (Do  not  use  logarithms.) 

41.  A  railroad  embankment  is  9  ft.  high.     The  length  of  the  slope 
of  the  embankment  on  each  side  is  14  ft.     Find  the  angle  which  the 
slope  makes  with  the  horizontal,  and  also  find  the  width  of  the  embank- 
ment at  the  base  if  the  top  is  8  ft.  wide. 

42.  Given   the   triangle    ABC,    whose    sides    are    AB  =  87.6    yd., 
AO=  112.7  yd.,  and  BC  =121.6  yd.     A  point  D  is  taken  on  the  line 
AC  produced  through  C,  so  that  the  angle  BDC  is  18°  37'  48"  [18.63°]. 
Find  the  distance  DC. 

43.  The  area  of  a  triangle  is  3  acres  and  two  of  its  sides  are  92.6 
and  26.72  rd.     Find  the  angle  between  these  sides. 

44.  A  shooting  star  is  observed  at  two  places  200  mi.  apart  on  the 
earth's  surface ;  the  angle  of  elevation  of  the  star  at  one  station  is  27° 
and  at  the  other  is  63°,  the  star  being  in  the  same  plane  with  the  two 
stations  and  the  center  of  the  earth.     Taking  the  radius  of  the  earth  as 
3956  mi.  find  the  height  of  the  shooting  star  above  the  earth's  surface 
and  hence  the  height  of  the  earth's  atmosphere.     (What  is  a  shooting 
star  ?     What  causes  its  light  ?) 

45.  Show  how   to  solve  each  of  the  cases  in  oblique  triangles  by 
dividing  the  oblique  triangle  into  right  triangles  and  using  the  methods 
of   solving  right   triangles   given   in   Chapter   III.     Why  do  we  not 
ordinarily  use  this  method  of  solving  oblique  triangles  ? 

46.  Make  up  (or  collect)  all  the  different  examples  you  can  showing 
practical  applications  of  trigonometry,  each  example  being  distinct  in 
principle  or  in  field  of  application  from  the  other  examples. 


CHAPTER   VIII 

CIRCULAR   MEASURE.     GRAPHS   OF   TRIGONOMETRIC 

FUNCTIONS 

100.   Radians,  or    the    Circular    Measure   of  Angles.     The 

method  of  measuring  angles  by  taking  a  right  angle  as  the 
unit,  dividing  the  right  angle  into  90  degrees,  dividing  each 
degree  into  60  minutes,  etc.,  is  called  the  sexagesimal  method 
and  originated  in  Babylonia  (see  Art.  127)  in  very  early 
times.  It  continues  to  be  generally  used  in  spite  of  its 
awkwardness  because  of  the  extensive  tables  and  large 
number  of  results  stated  in  terms  of  it  which  have  been 
accumulated. 

However,  the  advantages  of  the  decimal  division  of  any 
unit  are  so  great  that  it  is  a  growing  custom  to  divide  the 
degree  of  angle  into  tenths  and  hundredths  instead  of 
minutes  and  seconds  (see  many  examples  in  this  book). 

Also  within  the  past  century  it  has  become  customary  in 
many  kinds  of  work  (especially  algebraic  or  theoretic  work) 

to  use  a  unit  of  angle  different  from 
the  right  angle,  called  the  radian,  and 
to  divide  this  unit  decimally. 

A  radian  is  the  angle  which,  when 
its  vertex  is  placed  at  the  center  of  a 
circle,  intercepts  an  arc  equal  to  the 
radius  of  the  circle. 

FlQ  81  Thus  if  the  arc  AC  (Fig.  8)  equals 

the  radius  AB,   the   angle   ABC  is  a 

radian,  or  the  unit  angle  in  the  so-called  circular  method  of 
measuring  angles. 

142 


t 


CIRCULAR  MEASURE  143 

Hence,  to  determine  the  number  of  radians  in  an  angle 
whose    arc    and   radius  are  given,  we 
have  the  relation 

no.  of  radians  in  an  angle  =  -   , 

radius 

denoting  the  number  of  radians  in  an 
angle  by  />,  the  subtended  arc  by  a,  and 

the  radius  of  the  circle  by  R,  p=  —  . 

R 

Ex.  1.    Find  the  number  of  radians  FlG-  82> 

in  an  angle  AOB  whose  arc  is  13  and  radius  5. 
We  have,  Z  AOB  =  -1/  =  2.6  radians,  Ans. 

From  the  above  relation  it  follows  that 

Any  two  of  the  three  quantities,  number  of  radians  in  an 
angle,  arc,  and  radius,  being  given,  the  other  may  be  found. 

Ex.  2.    An  angle  containing  2.4  radians  subtends  an  arc 
14  in.  long.     Find  the  radius. 

^  Substituting  for  p  and  a  in  the  formula  p  =  — , 

R 

2.4=  it^Hi.  .-.  R  =  *ilE:  =  5.83+  in.,  Ans. 

R  2.4 

101.   I.  Converting  Degrees  into  Radians. 

The  number  of  radians  about  a  point  in  a  plane 
_  circumference 
radius 

_277#_9 

~R~' 

,.  3600  =  277,  or  6.2832  radians.  ^  ^  Q  ^  ^.^ 

180°  =  77,  or  3.1416  radians. 

•90°  =  |,  or  1.5708  radians.    30°  =  |,  or  0.5236  radians. 

60°  =  ^,  or  1.0472  radians.      I°  =  T^>  or  -01745  radians. 
3  180 


144  TRIGONOMETRY 

Hence  to  convert  degrees  into  radians 

Multiply  the  given  number  of  degrees  by  -   -  (or  by  .01745+). 

loU 

Ex.  1.    How  many  radians  in  26°  17'  36"? 

26°17'36"  =  26.293+° 

=  (26.293+)(.01745)  radians. 
=  0.45882+  radians,  Ans. 

Ex.  2.    Simplify  sin  (|  +  x). 

sinf  -  +  x  ]  =  sin  ^  cos  x  +  cos- sin  x  (Art.  66) 

\6       J  b  6 

=  i  cos  x  + i  V3  sin  x,  Ans.  (Art.  33) 

Where  the  meaning  is  evident  from  the  context,  it  is  customary  to 
abbreviate  "TT  radians"  into  "IT."  Thus  also  we  abbreviate  "sin- 
radians"  into  "sin-"  and  similarly  for  other  expressions. 

102.    II.    Converting  Radians  into  Degrees. 

Since  2  TT  radians  =360° 

180° 
1  radian  =  -    — , 

TT 

or  1  radian  =  57.29579+° 

=  57°  IT  45" 
=  206265". 

Hence  to  convert  radians  into  degrees 

180° 
Multiply  the  given  number  of  radians  by  -      -  (or  57.3°-). 

TT 

Ex.    Convert  2.5  radians  into  degrees,  minutes,  and  seconds. 

2.5  radians  =  2.5  x  (57.2958°-) 
=  143.2395° 
=  143°  14'  22",  Ans. 

9  Hence,  if  the  number  of  degrees  in  an  angle  be  denoted 
by  A,  the  number  of  radians  in  it  by  />,  etc.,  any  two  of  the 


CIRCULAR   MEASURE  145 

four  quantities  A,  p,  a,  R  being  given  (provided  one  of 
them  is  a  or  R),  the  other  two  may  be  found  by  substitution 
of  the  two  given  quantities  in  the  two  equations 

a  /180 

=  -  and 


103.  The  solution  of  a  right  triangle  containing  an  angle 
less  than  2°  may  often  be  conveniently  effected  by  the  use 
of  radians.  For  the  sine  or  tangent  of  a  small  angle  may  be 
taken  as  equivalent  to  the  number  of  radians  in  the  angle 
(i.e.  the  circular  measure  of  the  angle)  without  appreciable 
error  (see  Art.  115). 

Thus  sin  A  =  A  (in  radians)  when  A  is  a  small  angle,  is  an  ap- 
proximation frequently  used  in  Physics,  and  the  result  is  accurate  to 
within  the  probable  degree  of  error  in  measurement. 

Ex.    If  a  railroad  track  has  a  rise  of  1  ft.  in  every  2000  ft. 
in  its  length,  what  angle  does  it  make  with  the  horizontal  ? 
Denoting  the  required  angle  by  A, 

sin  A  =  —  —  =  no.  radians  in  A  approximately. 
2000 

x  206265"  =  103+  "  =  V  43",   Ans. 


EXERCISE   42 


1.  Reduce   the   following   angles   to   circular   measure,   expressing 
the  results  as  fractions  of  TT  : 

30°,  135°,  60°,  90°,  210°,  270°,  225°,  72°,  315°. 

2.  Express  the  following  angles  in  degrees  : 

TT         7T          7T          2  TT         4  7T          3  7T          7  TT         8  77 

6'    4'    3'    T'    IT'    ~6~'    T'     15* 

3.  What  decimal  part  of  a  radian  is  1°  ?   16"  ?  2'  15"  ?  5°  14'  ? 

4.  How  many  degrees   (minutes  and  seconds)  in  2  radians  ?    3.2 
radians  ?    .003  radians  ? 

5.  A  circle  has  a  radius  of   14   inches.     How   many    radians  are 
there  in  an  angle  at  the  center  subtended  by  an  arc  21  in.  long  ?  By 
an  arc  7  in.  long  ? 


146 


TRIGONOMETRY 


6.  In  a  circle  of  radius  R,  an  arc  3  ft.  6  in.  subtends  an  angle  of 
1.5  radians.     Find  It. 

7.  One   angle   of  a  triangle  is  30°,  and  the  circular   measure  of 
another  angle  is  1.5  radians.     Find  the  third  angle  in  degrees.     Also 
in  radians. 

8.  The  difference  between  two  angles  is  -  and  their  sum  is  110°. 
Find  the  angles  in  degrees ;  in  radians. 

9.  Find  both  in  radians  and  degrees  the  complement  and  supple- 
ment of  the  following  angles : 

—       —       "*       JL      **7r 

6'    3'    4'    9'    18* 

10.  Write  out  the  trigonometric  ratios  of  the  following  angles : 

7T          7T          7T          7T          3  7T          7  7T          7  7T 

6'    3'    4'    2'    T>    IP     T' 

11.  How  many  radians  in  an  angle  whose  arc  is  12  and  radius  10  ? 
How  many  degrees  ? 

12.  Show  that  sin  (x  +  1  ?r)  +  sin  (x  —  %  TT)  =  sin  x. 

Supply  the  two  missing  quantities  in  each  of  the  following : 


13 
14 
15 
16 
17 

p 

a 

R 

A 

2.5 
.25 

10  in. 
12ft. 
100 

50  in. 
1  ft.  6  in. 
42  in. 

1°30' 
37° 

18.  If  a  railroad  track  has  a  rise  of  1  ft.  in  750  ft.,  what  angle  does 
the  track  make  with  the  horizontal  ? 

19.  If  a  railroad  makes  an  angle  of  1°  30'  with  the  horizontal,  what 
is  its  rise  in  one  half  mile  ? 

20.  An  irrigating  ditch  should  have  a  fall  of  at  least  \  in.  per  rod. 
What  angle  does  the  bottom  of  the  ditch  make  with  the  horizontal  ? 

21.  If  the  moon  is  at  a  distance  of  240,000  mi.  from  the  earth  and 
the  radius  of  the  moon  subtends  an  angle  of  16'  as  seen  from  the  earth, 
what  is  the  radius  of  the  moon  in  miles  ? 

22.  If  the  sun  is  at  a  distance  of  92,800,000  mi.  from  the  earth,  and 
the  diameter  of   the  sun  subtends  an  angle  of  32.4'  as  viewed  from 
the  earth,  what  is  the  radius  of  the  sun  in  miles  ? 

23.  The  planet  Mars  has  a  diameter  of  4200  miles.     When  Mars  is 
nearest  the  earth,  its  diameter  subtends  an  angle  of  24.5"  as  seen  from 


CIRCULAR  MEASURE  147 

the  earth.      What  is  the  distance  of   Mars  from  the  earth  at  such  a 
time? 

24.  Find  the  numerical  value  of  3  sin  -  -4  cos  ^  tan  -  +  cot  ^  • 

25.  Make   up   two  practical   problems   in   each  of   which   a  right 
triangle  is  solved  by  the  use  of  radians  as  in  Exs.  17-21. 

We  shall  now  illustrate  the  use  of  radians,  or  the  circular 
measure  of  angles,  (1)  in  tracing  the  graphs  of  trigonometric 
functions,  (2)  in  solving  trigonometric  equations. 

GRAPHS   OF   TRIGONOMETRIC   FUNCTIONS 

104.  Graph  of  sin  oc.  To  form  what  is  called  the  graph 
of  sin  x  use  the  equation  y  =  sin  x  and  also  a  pair  of  rectan- 
gular axes  (see  Art.  54).  In  the  equation  y  =  sin  x,  let  x  have 
convenient  successive  values  and  find  the  corresponding  values 
of  y.  Lay  off  each  corresponding  pair  of  values  of  x  and  y  as 
the  abscissa  and  ordinate  of  a  point.  Draw  a  continuous  curve 
through  the  terminal  points  thus  located. 

It  is  usually  convenient  to  make  the  scale  of  the  drawing 
such  that  a  unit  space  of  the  cross-section  paper  stands 

for  \  or  .5236+. 

6 

Thus,  if  we  desire  to  make  a  graph  of  y  =  sin  x  we  may  take  the 
following  corresponding  values  of  x  and  y  : 


x  =  ir,  y  =  0,  etc.  $  =  —  IT,  y  =  0,  etc. 


148 


TRIGONOMETRY 


Using  these  results,  the  curve  AOBCDE  (Fig.  83)  is  obtained 
as  the  graph  of  sin  x.  Such  a  figure  shows  at  a  glance  the 
changes  in  the  values  of  sin  x  as  x  changes  in  value. 


FIG.  83. 


105.  Graphs  of  Other  Trigonometric  Functions.  By  treat- 
ing the  equations  y  =  cos  x,  y  =  tan  x,  y  =  sec  x9  etc,  simi- 
larly, the  graphs  of  the  other  trigonometric  functions  may 
be  constructed. 


It  is  important  to  observe  in  constructing  the  graph  of 

tanz,  that,  as  x  =  ^9  y  =  either  +  GC  or  -  oc.       For  as  we 

2i 

proceed  from  x  =  0  and  make  x  =  o",  y  =  +  GC;  but  as  we 
proceed  from  x=  TT  and  make  x  ==  ^,  y  ==  —  oc.     Hence  we 


CIRCULAR   MEASURE  149 

obtain  as  part  of  the  graph  of  tan  x  the  curve  AOB,  CO'D  of 

Fig.  84. 

EXERCISE  43 
Graph  each  of  the  following  : 

1.  y  =  sin  x.  9.  y  =  tan  1  x. 

2.  y  =  cos  x.  10.  y  =  sin  x  -f  cos  x. 

3.  y  =  tan  x.  11.  y  ==  sin  x  —  cos  #• 

4.  y  =  cot  a?.  12.  y  =  Vsin  a;. 

5.  y  =  sec  #.  13.  y  =  sin2  #.                      % 

6.  y  =  esc  #.  14.  y  =  1  +  sin  x. 

7.  y  =  sin  i  x.  15.  y  =  l  —  cos  a. 

8.  y  =  sin  2  x.  16.  y  = 


106.  Solutions  of  Trigonometric  Equations.     Answers  not 
greater  than  360°,  i.e.  than  2  TT  radians. 

Ex.  1.     Find  the  values  of  x  less  than  2  TT  radians  which 
shall  satisfy  the  equation  sin  x  =  J. 
Since  sin  30°  =  \,  and  also  sin  150°  =  |, 

x  —  -   or  —  ^  radians.  ^4ns. 
6  6 

Ex.  2.  Solve  4  cos  a:  —  3  sec  #  =  0  for  values  of  x  less 
than  2  TT. 

Q 

4  cos  a;  --   —  =  0. 

cos  x 

4  cos2  x  -  3  =  0. 

cos  x  =  ±1  V3. 
Hence,  *  =  30°,  150°,  210°,  330°, 

or  *  =  I',1T'   T'    ^  mdiaUS'  ^S* 

107.  Answers  Unlimited. 

Ex.  1.     Solve  the  equation  cos  x  =  \. 

One  value  of  x  is  60°  and  another  value  is  -  60°.  But  if  360°  be 
added  to  or  subtracted  from  the  value  of  an  angle,  the  value  of  the 
function  is  unchanged. 


150  TRIGONOMETRY 

Hence,  x  =  2  mr  ±  ^   radians,  where  n  is  zero  or  any  positive   or 
negative  integer. 

Ex.  2.     Solve  the  equation  sin  x  —  esc  x  +  f  =  0. 

Solving  the  equation,  we  obtain, 

sin  x  =  —  2,  i. 

Since  the  sine  of  an  angle  cannot  be  greater  than  1,  no  angle  corre- 
sponds to  the  value  —  2. 

For    ,  sin  x  =  i, 

£,  (2 n +  !>-£,  Ans. 


EXERCISE  44 

Solve  each  of  the  following   equations,  expressing  the  answers  in 
radians,  by  use  of  TT. 

1.  cot2  6  =  -  3.  12.  Cot  g  + 1  =  cos  2x. 

cot  x  —  1 

2.  tan2  0  =  3.  13.  2  sin2  a;  —  sin  x  =  sin  2  x— cos  x. 

3.  cot2  0=1.  14.  cos  2  x  -f-  cos  a?  =  0. 

4.  sin2  0  =  f .  15.  tan  (45°  +  a;)  +  tan  (45°  -  x) = 4. 

5.  cot  0  =  2  cos  0.  16.  2  esc2  a;  —  V3  cot  x  =  5. 

6.  cos  0  +  sec  0  =  f.  17.  sin  3  x  =  sin  5  x  +  sin  a;. 

7.  3  sin2  x  -h  cos2  a?  =  f .  18.  cos  3  a;  +  cos  a?  =  cos  2  #. 

8.  3  cot2  x  -f  tan2  a?  =  4.                  19.  sin  5  x  —  sin  x  =  cos  3  x. 

9.  cos  x  =  sin  2  a?.  20.  cos  3  x  —  cos  a;  =  —  sin  2  a?. 

10.  cos  2  x  +  sin  a?  =  4  sin2  .T.         21.  sin  o  cc  +  sin  3  x  +  sin  #  =  0. 

11.  sin  2  #  =  tan2  x.  22.  cos  5  x  +  cos  3  x  +  cos  a:  =  0. 

108.    Simultaneous  Trigonometric  Equations. 

Ex.  1.     Solve  x  sin  y  =  a ',  - 

for  x  and 
x  cos  w  = 


Dividing  the  first  equation  by  the  second, 

tan  y  =  -'      .\  y  =  /-  whose  tan  is  -,  Ans. 
b  o 

(For  a  briefer  way  of  expressing  this  result  see  Chapter  IX.) 


CIRCULAR  MEASURE  151 

From   this  result  the  value  of  y  may  be  obtained.     When  y  is  known 
x  can  be  obtained  from  either  of  the  original  equations. 


OTX  = 


sin  y  cos  y 

Ex.  2.     Solve  for  x  and  y  the  equations, 

x  cos  A  +  y  sin  ^4  =  a (1) 


x  sin  A  -  y  cos  A  =  b (2) 

Multiply  equation  (1)  by  cos  A,  then 

x  cos2  A  +  y  sin  A  cos  A=a  cos  ^4 (3) 

Multiply  equation  (2)  by  sin  A,  then 

x  sin2  A  —  y  sin  ^4.  cos  A  =  b  sin  A (4) 

Add  (3)  and  (4),  using  the  fact  that  sin2  ^4  +  cos2JL=  1, 
then  x  =  a  cos  A  -\-b  sin  A, 


and  similarly,  y=  a  sin  A—  b  cos  A.  ! 


EXERCISE  45 

Solve  for  x  and  0,  or  for  x  and  ?/  : 

f  x  cos  0  =  86.65,  fa  tan  0  =  816.95, 

{  x  sin  0=50.  {  x  sin  0  =  426.3. 

f  x  sin  0  =  118.96,  f  x  sin  ?/  =  4, 

{a  cos  0  =  160.78.  4'    |>cosy  =  8. 


f  «  sin  30°  +  y  cos  45°  =  53.28, 
5*    I  x  cos  30°  +  y  sin  45°  =  71.58. 

f  x  sin  48°  -f  y  cos  19°  =  2634.1  , 
5*  »  (  x  cos  48°  +  y  sin  19°  =  1320.3. 

r  sin  x  +  sin  y  =  1.573,     [Use  Art.  71.] 
\  cos  x  +  cos  y  =  1.207. 


f  sin  #  —  sin  y  —  .2154, 
\  cos  x  —  cos  y=—  .1231. 

(  x  sin  (0-21.5°)  =  771.1, 
\aj  cos  (0-32.5°)  =  766. 

f  x  cos  J.—  y  sin  A  =  a, 
{  x  sin  ^1  +  y  cos  .4  =  6. 


V    CHAPTER   IX 
INVERSE   TRIGONOMETRIC   FUNCTIONS 

109.  Anti-sine.     If  y  is  an  angle  and  x  its  sine,  the  relation 

between  x  and  y  may  be  expressed  in 
either  of  two  ways  : 

(1)  x  =  siny,  or 

(2)  y  =  sin"1  x,  which  reads  "  y  is  the 
angle  whose  sine  is  x"  or  " y  is  the  anti- 
sine  of  xr 

One  or  the  other  of  methods  (1)  or  (2)  is  used  according 
as  the  angle,  or  its  sine,  has  the  leading  place  in  the  discus- 
sion. Thus  if  the  angle,  or  y,  is  more  prominent,  x  =  sin  y 
is  used;  but  if  the  sine,  x,  is  more  prominent,  y  =  sm~1x  is 
used. 

The  pupil  should  carefully  discriminate  between  sin"1^  and  the  —  1 

power  of  sin  x.     The  latter  is  expressed  thus,  (sin  x)~\     Thus, = 

sin  a; 

(sin  a?)"1,  and  not  sin'1  x.     But  (sin  x)~2  may  be  written  sin~2x. 

110.  Other  Anti-trigonometric  Functions.    Similarly  cos"1  x 
means  "  the  angle  whose  cosine  is  x  "  ;  tan"1  x  means  "  the 
angle  whose  tangent  is  x."     Let  the  pupil  state  the  meaning 
of  cotrl#,  csc"1^,  vers"1^. 

It  is  evident  that  sin  (sin"1  x)  =  x,  since  the  sine  of  the  angle  whose 
sine  is  x  must  be  x.  Similarly  cos  (cos"1  a;)  =  x,  etc. 

Hence  there  is  a  similarity  in  form  between  a(a~l)x  =  x,  and 
sin  (sin-1  x)  =  x.  It  is  because  of  this  similarity  that  the  system  of 
symbols  described  above  is  used  to  express  the  anti-trigonometric 
functions. 

152 


INVERSE  TRIGONOMETRIC   FUNCTIONS 


153 


A  much  better  symbolism  for  "y  equals  the  angle  whose  sine  is  x" 
would  seem  to  be  "y  =  Zsmx,"  and  if  the  pupil  has  difficulty  in 
grasping  the  principles  of  this  chapter,  it  may  be  well  for  him  to  use 
this  latter  method  of  writing  inverse  functions  till  he  becomes  familiar 
with  their  nature. 

111.  Values  of  Inverse  Trigonometric  Functions.  The 
direct  and  inverse  trigonometric  functions  have  an  important 
difference  with  reference  to  the  number  of  values  which 
satisfy  them. 

Thus,  if  y  =  sin  30°,  y  has  a  single  value,  J;  but  if  x  — 
sin"1  J-,  x  can  have  an  indefinite  number  of  values,  viz. :  30°, 
150*,  390°,  510°,  etc.;  or 

x=2nw+%,  (2  n +  ]>-£•    (See  Art.  107,  Ex.  2.) 
o  6 

For  many  purposes  it  is  customary  to  limit  the  values  of 
an  inverse  circular  function  to  the  smallest  value  that  will 
satisfy  a  given  expression. 

Thus,  if  0  =  tan'1  1,  we  take  0=  45°. 


112.    Given  an   Anti-trigonometric   Function,  to   find   the 
other  Related  Functions. 

Ex.  1.  Given  6  =  tan"1  f ,  find  sin  9 ; 
that  is,  find  sin  (tan"1  J-). 

6  =  tan"1 1^  may  be  converted  into  the  form 
tan  0  =  f  for  which  a  diagram  may  be  con- 
structed (Fig.  86). 


.-.  sin  (tan-1  f)  =  ^ Vl3  Ans. 

Ex.  2.     Find  sin  2(cos-!  J). 
Let  x  be  the  angle  whose  cosine  is  i. 
Then  cos  x  =  £,  sin  x  =  Vl  — 

.-.   sin.  2  a;  =  2  sin  x  cos  x  = 
Hence,  sin  2(cos~1  ^)  =  | V2, 


3 
FIG.  86. 


|V2. 

=  |V2. 


154 


TRIGONOMETRY 


Ex.  3.     If  0  =  tan  l  a,  express  the  direct  and  inverse  func- 
tions of  6  in  terms  of  a. 

tan  6  =  a,  hence  0  =  tan"1  a. 

1 

a 
0  =  sec~1Vl  4-  a2. 

1 


cot  0  =  -, 
a 


sec  6  =  Vl  4-  a*, 


cos     = 


1 


Vl  4-  as 


1 

FIG.  87. 


sin  0  = 


CSC       = 


=  sin 


VI  +  a2 
a 


VI  4-  a 


Ordinarily  only  the  positive  value  of  each  radical  is  used. 

113.    Inverse  Trigonometric  Functions  of  Two  Angles. 

Ex.  1.     Find  sin  (sin'1  \  4  cos'1  -f). 

Let  x  =  sin"1 1. 
.-.  sin  x  =  ^-, 

cos  a?  =^V3. 

Let  i/  =  cos"1 f . 

cos  y  =  |, 
.-.  sin  y  = 


\y 


FIG.  88. 


2 
FIG.  89. 


Then  sin  (sin"1  1  4-  cos"1  1)  =  sin  (x  -h  y)  =  sin  #  cos  y  +  cos  x  sin  y 


=  J(2+V15), 

Ex.  2.    Prove  that  sin"1  a  4-  cos"1  a  =  ^) 
Using  the  method  of  Ex.  1,  show  that 

sin  (sin"1  a  4-  cos"1  a)  =  1  =  sin  f  . 

Ex.  3.    Show  that  tan-1  a  +  tan'1  6  -  tan-1  £±^-. 

l-ab 
x  =  tan"1  a. 


Let 


But 


.-.  a  =  tan  x, 

y  =  tan"1 

'.*.  b  =  tan  ?/. 


i 
FIG.  90. 


1 
FIG.  91. 


/          N       tan  x  4-  tan  y 
tan  (x  4-  y)  =  —  — —• 

'      1  —  tan  x  tan  y 


INVERSE   TRIGONOMETRIC   FUNCTIONS  155 

.-.  tan  (tan-1  a  +  tan-1  b)  =  ^\t  or  tan-1  a  +  tan-1  6  =  tan'1  «+A. 

1  —  a&  1  —  ab 

114.  Solution  of  Trigonometric  Equations  by  Use  of  In- 
verse Trigonometric  Functions.  It  is  sometimes  useful  to 
express  the  answer  obtained  by  solving  a  trigonometric  equa- 
tion in  terms  of  an  inverse  function. 

Ex.    Solve  6  cos2  x  —  cos  x  =  2. 

Factoring,    (2  cos  x  +  1)(3  cos  x  —  2)  =  0. 
.-.  coscc  =  —  !,  f. 

.-.  «  =  cos"1  (—  i),  cos"1 1,   Ans. 

EXERCISE  46 

If  the  pupil  has  any  difficulty  in  grasping  any  one  of  the  following 
problems,  it  will  be  well  for  him  to  translate  the  symbols  of  the 
problem  into  general  language  before  attempting  the  solution.  Thus 
Ex.  2  would  read  "  find  the  cosine  of  the  angle  whose  cotangent  is  }," 
and  might  be  written  in  the  form. "find  cosZ  cotf  "  (see  Art.  110). 

Express  the  following  angles  first  in  degrees  and  then  in  radians : 

1.  cos-^VJ?,   tan-'VS,    sin-1!,    sec^V^,    csc^fVS,    cot^VS* 
cos^i,    sec-1  2,   sin-^VS,  cot-^Va,  tan-1^  V3. 

Find  the  value  of : 

2.  cos  (cot-1  f).  8.  sin  (2  tan-1 3^). 

3.  tan  (sin-1  T%).  9.  cos  (2  sec'1  -1/)- 

4.  sec  (tan"1  -£%).  10.  sin(icos-1i). 

5.  sin  (cot-1  a).  11.  cot(|  tan-1-1/)- 

/          ft\  12.    sin  (3  sin-1!). 

6.  cot  (cos-1- ). 

b'  13.   sm  (sm-1 1  -  cos-1  f ). 

7.  tan  (2  sin^i).  14.    tan  (tan-1  2  +  cot"1  3). 
Show  that : 

15.    tan^i  +  tan-1^.  16.    tan- 


17.  sin-1  T8T  +  sin-1  f  =  sin"1  £J. 

18.  cos-1  f  +  cos-1  fV  =  cos-1  (-  f|). 

19.  tan-1  f  +  tan  T\  =  tan"1  JJ. 

20.  cot-1  a  +  cot-1  b  =  cot-1  (t6~1. 


156  TRIGONOMETRY 

Prove  that : 

21.  sin  (sin-1  f  +  cot"1 1)  =  1. 

22.  (cos"1  |f  +  tan-1  ^-)  =  sin"1  11J-- 

23.  sin  (2  tan-1  a?)  =    w  ^  g. 

1  T  «C 

24.  sin"1  #  =  cot"1 — . 

25.  cos"1  a  —  cos"1  b  =  cos~1  (ab  +  Vl  —  a2  —  b-  +  a262). 

26.  3  cos  -1  a;  =  cos"1  (4  3?  —  3  #). 

27.  3  sin  -1  x  =  sin-1  (3  x  -  4  or3). 

28.  tan-1  a -tan-1  6  =  -^^. 


29.  sin"1  a  +  sin'1  6  =  eo8~l(vl  —  a2  —  62  +  a262  —  ab). 

Express  the  value  of  each  of  the  following  in  its  most  general  form : 

30.  sin-1!  35.   cos-^VS. 

31.  tan-1  £  V3.  36.   tan"1^. 

32.  cos-^V^.  37.    cot-1  V3. 

33.  cot-1  £  VS.  38.    sec^V^. 

34.  sin-1  ^  VS.  39.    sin-^-l). 


40.  Prove  that  tan  (2  tan"1  a)  = 

41.  Prove  sin  (2  tan-1  a)  = 


1-a2 


.  a2 

42.  If  cos"1  x  —  2  cos"1  a?,  find  x. 

43.  Express  the  following  angles  in  the  inverse  notation :   30°,  60°, 
90°,  45°,  0°;  n!80°,  n90°. 

Can  each  of  these  angles  be  expressed  in  more  than  one  way  in  the 
inverse  notation  ? 

44.  Who  first,  and  at  what  time,  brought  inverse  circular  functions 
into  use  in  their  present  form  (see  p.  173)  ? 

45.  :At  what  time  did  the  circular  method  of  measuring  angles  come 
into  use  (see  p.  167)  ? 


CHAPTER   X 
COMPUTATION  OF  TABLES 

TRIGONOMETRIC   SERIES 


115.    Limiting  values  of 


05 


and 


It  is  important 


to  determine  the  values  which  - '- —  and    anx  approach  when 

x  x 

x  =  Q,  x  being  the  value  of  an  angle  expressed   in  circular 
measure  (radians). 

Take  any  angle  AOP  (Fig.  92) 
less  than  90°  and  denote  it  by  x ; 
construct  the  angle  AOP'  equal  to  o 
AOP,  and  draw  the  tangents  PT 
and  P/  T.  These  tangents  will  meet 
at  I  on  OA  produced.  Draw  PP' . 

Then  OT  is  _L  to  PP'  at  its  middle 
point  M. 

By  geometry,     arc  PP'  >  chord  PP' ; 
also  McPP'<PT+P'T. 

Hence  arc  PA  >  PM,  and  arc  PA  <PT. 


arc  PA     PM       ,  wcPA 

-oj^>op>™d-oi^* 

.'.  x>  sin  x,  and  x  <  tan  x. 
x  1 


PT 
OP' 


sin 


>  1,  and 


sin  x 
s\ux 


cos  x 


x 


157 


158  TRIGONOMETRY 

Ql  Yl    *y*  ^11  Yl   'V1 

As  x  =  0,  cos  x  =  1,  hence  -    —  =  1,  since  -      -  lies  between 
cos  x  and  1. 

•  AT    'A.  /'sin  x\      -, 
Hence  as  x  ==  0.  limit  (  —     • )  =  1 . 

>•    x   / 

This  result  may  also  be  stated  thus,  as  x  =  0,  sin  x  =  x. 

A  T      tan  x       sin  x        /^sin  xA  /    1 
Also  -  _  =  (-_)[- 

X  X  COS  X         >•     X     /   ^COS 


.  A    sin  x  .  -,         ,      1      .1 
But  as  x  =  0,  -     —  =  1,  and  -     —  =  -  or  1. 

X  COS  X       1 


Hence  =lx  1,  or  1. 


x 


AT    -j.  -, 

Or.  as  x  =  0.  limit  (        -  )  =  1. 


x 


arc  A.  P 
Since  the   number  of    radians  in  x=  —  —  —  ,  it  follows 


that  as  the  angle  x  =  0,  the  number  of  radians  in  x  =  sin  x, 
and  also  =  tan  x. 

In  practical   work,  when   x<2°,  sinx  and  tanx  may   be 
taken  as  =  p  without  appreciable  error. 

116.  Computation  of  the  Tables  of  Trigonometric  Func- 
tions. Since,  as  x  =  0,  sinx  and  x  approach  equality  (Art. 
115),  the  circular  measure  of  a  small  angle  is  the  same  as  the 
sine  of  that  angle  to  a  large  number  of  decimal  places.  By 
the  use  of  methods  which  are  beyond  the  scope  of  this  book 
it  is  found  that  the  value  of  sin  1'  and  the  circular  measure 
of  r  coincide  for  the  first  fourteen  decimal  places.  Hence 
in  constructing  tables  which  are  to  be  correct  for  the  first 
five  decimal  places,  there  will  be  no  error  in  taking 
sin  1'  =  1'  (in  radians). 

But,  by  Art.  101, 

I'  =  3'141592+  radians  =  .0002908882+  radians. 
180  x  60 

Hence  sin  1'  =  .0002908882+. 


COMPUTATION   OF   TABLES  159 


But       cos  r  =  Vl  -  sin2  1'  =  Vl  -  (.0002908882+)2 

=  .9999999577+. 

sin  2'=  2  sin  V  cos  r  =  2  x  (.0002909-)(.9999999577+) 
=  .000582+. 

sin  3'  =  sin  (2'  + 1')  =  sin  2'  cos  1'  +  cos  2'  sin  r. 

From  this  the  value  of  sin  3'  may  be  computed. 

In  like  manner  the  sines  of  all  angles  less  than  90°  may 
be  obtained. 

The  cosines  of  these  angles  may  be  obtained  similarly,  or 
by  use  of  the  formula  cos  x  =  sin  (90°  —  x). 

The  tangents  of  these  angles  may  be  computed  by  the  use 

Q-I  i"*     /y» 

of  the  formula  tanx=-    — .     To  obtain  the  cotangents,  the 

cosx 

formula  cot  x  •=  tan  (90°  —  x)  may  be  used. 

The  above  method  of  computing  sines  and  cosines  may  be 
abbreviated  thus : 

sin  (x  4-  y)  +  sin  (x  —  y)  =  2  sin  x  cos  y.        (Art.  71) 
Let  x  =  a  +  2  b,  and  y=b.     Then,  by  substitution, 
sin  (a  +  3  b)  -f-  sin  (a  +  b)  =•  2  sin  (a  +  2  b)  cos  b. 

Whence 

sin  (a  -f  3  b)  =  2  sin  (a  +  2  b)  cos  6  -  sin  (a  +  6).  .     .  (1) 

In  like  manner, 

cos  (a  -f  3  b)  =  2  cos  (a  +  2  &)  cos  6  -  cos  (a  +  &).        .  (2) 

Let  6=  r  in  (1)  and  (2). 

sin(a  +  3/)  =  2sin(a  +  2/)cosr-sin(a+r).  .     .   (3) 
cos(a  +  3')  =  2  cos  (a  +  2')  cos  l'-cos  (a  +  r).  .     .   (4) 

Letting  a  =  —  T,  0,  1',  2',  ...  in  succession,  we  obtain 

from<3)  sin  2'  =  2  sin  1' cos  r. 

sin  3'  =  2  sin  2'  cos  r  -  sin  1'. 
sin  4'  =  2  sin  3'  cos  1'  -  sin  2X,  etc. 


160  TRIGONOMETRY 

Similarly  from  (4), 

cos2'=2cosl'-l. 

cos  3'  =  2  cos  2'  cos  V  —  cos  Y. 

cos  4'  =  2  cos  3'  cos  Y  —  cos  2',  etc. 

117.  Computation  by  the  Use  of  Series.  The  computation 
of  the  numerical  values  of  the  trigonometric  functions  is, 
however,  performed  much  more  expeditiously  by  the  use  of 
certain  trigonometric  series  than  by  the  above  method.  The 
demonstration  of  these  series  lies  beyond  the  scope  of  this 
work.  The  series  are  as  follows  : 

X3    ,    X5        X*    , 

SmX  =  *-+-  • 


^  x*      2  yf      17  x3  , 
=  z+-  +  -—  +  ——  +  -  -  - 
6       15        olo 

The  student  is  aided  in  recalling  these  series  by  the  fact 
that  sin  (  —  x)  =  —  sin  x  (Art.  63)  ;  hence  sin  x  must  equal  a 
series  composed  of  odd  powers  of  x.  The  same  is  true  of 
tan  x.  But  since  cos  (  —  x)  =  cos  x,  cos  x  must  equal  a  series 
composed  of  even  powers  of  x. 

118.  Analytical  Trigonometry.  Theory  of  Functions. 
When  trigonometry  is  treated  in  the  way  indicated  in  cer- 
tain preceding  articles,  it  ceases  to  be  merely  an  instrument 
for  solving  triangles  and  becomes  the  theory  of  quantities 
varying  in  certain  periodic  or  rhythmic  ways. 

Also  by  the  use  of  the  so-called  imaginary  quantities,  the 
subject  of  trigonometry  is  still  further  extended.  Thus,  for 
instance,  denoting  V  —  1  by  the  symbol  i,  it  is  shown  that 

(cos  x  +  i  sin  x)n  =  cos  nx  +  i  sin  nx 
(called  De  Moivre's  Theorem). 


COMPUTATION   OF   TABLES  161 

By  the  aid  of  this  theorem  and  similar  principles,  trigo- 
nometry gains  much  additional  power.  This  branch  of  the 
subject  is  termed  analytical  trigonometry  (though  it  is  some- 
times treated  as  a  part  of  higher  algebra). 

When  trigonometry  is  extended  in  these  various  ways,  it  is 
also  looked  upon  as  a  part  of  the  larger  subject,  the  theory 
of  functions. 

EXERCISE  47 

* 

1.  By  use  of  De  Moivre's  Theorem  obtain  the  formulas  for  sin  3  a/- 
and cos  3  x, 

By  use  of  this  theorem  we  obtain 

(cos  x  +  i  sin  a;)3  =  cos  3  x  +  i  sin  3  x. 
But 

(cos  x  +  i  sin  x)3  =  cos3  x  -f-  3  i  sin  x  cos2  x  -\-  3  i2  sin2  x  cos  x-\-  is  cos3  x. 
.'.  cos  3  x  -f-  i  sin  3  a;  =  cos3  x  —  3  sin2  x  cos  x  -\-  i  (3  cos2  x  sin  x  —  sin3  x). 

By  a  theorem  of  algebra,  in  an  identical  equation  containing  both 
real  and  imaginary  quantities,  the  sum  of  the  reals  in  one  member  is 
equal  to  the  sum  of  the  reals  in  the  other  member,  and  so  with  imagi- 
naries.  Hence, 

cos  3  x  =  cos3  x  —  3  sin2  x  cos  x  =  4  cos3  x  —  3  cos  x 
sin  3  x  =  3  cos2  x  sin  x  —  sin3  x  =  3  sin  x  —  4  sin3  x. 
In  like  manner,  by  De  Moivre's  Theorem,  prove : 
sin  4  x  =  2  sin  2  x  (1  —  2  sin2  a;), 


2 

1  cos  4  x  =  8  cos4  x  —  8  cos2  x  +  1. 

f  sin  5  a?  =  16  sin5  x  —  20  sin3  a;  -h  5  sin  x, 
3      I 

\  cos  5  a?  =  16  cos5  x  —  20  cos3  x  +  5  cos  a?.' 

4.  sin  7  x  —  7  sin  a;  —  56  sin3  x  +  112  sin5  a,*  —  64  sin7  x. 

7*0-1) 

5.  cos  nx  =  cosn x-  4  cosw-2 x  sin2 x 


2)(n  -  3) 

-  cos71"  x  sin  x 


1"4 


6.  sin  7ix  =  n  cos"-1  a;  sin  x  -  ^-  '   ^v         -'  cosn-3  x  sin3  a? 

n  (n  -  1)  (71  -  2)  (?i  -3)  (n  -  4)        n_5       .    6 

~W 

7.  tan2x  =  -^tana; 

1  —  tan2  x 

8.  Find  the  value  of  sin  225°  by  use  of  the  formula  for  sin  5  a;  in 
Ex.  3. 


CHAPTER   XI 
HISTORY  OF   TRIGONOMETRY 

119.  Epochs  in  the  History  of  Trigonometry.  The  begin- 
nings, or  germs,  of  Trigonometry  are  found  in  the  Rhind 
Papyrus,  now  preserved  in  the  British  Museum.  This  papy- 
rus, the  oldest  known  mathematical  document,  was  written 
by  a  scribe  named  Ahmes  about  1400  B.C.,  and  is  a  copy,  so 
the  writer  states,  of  a  more  ancient  work,  dating,  say, 
3000  B.C.,  or  several  centuries  before  the  time  of  Moses.  In 
dealing  with  pyramids,  Ahmes  makes  use  of  two  of  the 
trigonometrical  ratios,  viz. :  that  between  a  lateral  edge  of  a 
pyramid  and  diagonal  of  the  base,  corresponding  to  the  co- 
sine of  an  angle  ;  and  another  which  corresponds  to  the 
trigonometrical  tangent  of  the  angle  made  by  the  lateral 
face  of  a  pyramid  with  the  plane  of  the  base. 

This  use  of  ratios  is,  however,  too  crude  to  be  regarded  as 
scientific  trigonometry.  We  have  the  following  principal 
epochs  in  the  scientific  development  of  Trigonometry : 

1.  Greek  (at  Island  of  Rhodes  and  Alexandria),  150  B.C.- 
200  A.D. 

2.  Arab  (in  western  Asia  and  in  Spain),  650  A.D.-1200  A.D. 

3.  Hindoo,  450  A.D.-1100  A.D. 

4.  European,  1200  A.D.- 
We shall  also  find  the  three  following  principal  stages  in 

the  development  of  trigonometry: 

I.  (150  B.C.-1400  A.D.)  Spherical  Trigonometry  studied 
as  a  part  of  Astronomy,  with  incidental  use  of  Plane 
Trigonometry. 

162 


HISTORY   OF   TRIGONOMETRY  163 

II.  (1400  A.D.-1700  A.  D.)  Plane  and  Spherical  Trigonom- 
etry studied  as  a  part  of  Geometry. 

III.  (1700  A.D.-         )  Trigonometry  as  an   independent 
science. 

PRINCIPAL   MAKERS   OF   TRIGONOMETRY 

120.  Hipparchus.      The    founder    of    trigonometry    as    a 
science  was   Hipparchus,  a  Greek,  born  about  180   B.C.  in 
Bithyiiia  in  the  northern  part  of  Asia  Minor.     Hipparchus 
studied  at  Alexandria  and  afterward  retired  to  the  Island  of 
Rhodes,  where  he  did  his  principal  work.     He  was  primarily 
an  astronomer  and  determined,  for  instance,  the  length  of 
the  year  to  within  six  minutes.     He  created  trigonometry 
as  a  tool  or  aid  in  his  astronomical  work.     Hence  the  trigo- 
nometry used  by  him  was  almost  exclusively  spherical. 

121.  Ptolemy  (87  A.D.-165  A.D.).     The  next  great  name- 
in  the  history  of  trigonometry  is  that  of  Ptolemy,  also  a 
Greek.     He  lived  and  did  his  work  in  Egypt  at  Alexandria. 
Like  Hipparchus,  Ptolemy  was  primarily  an  astronomer  and 
used   trigonometry  merely   as    an    aid  in  his  astronomical 
investigations.     He  wrote  a  treatise  on  mathematical  and 
astronomical    topics,  now  known  as   the   Almagest,*  which 
was  the  standard  authority  in  astronomy  for  1200  years. 
The  Almagest  contains   thirteen    books,  the  first   of   which 
treats  mainly  of  trigonometry. 

122.  Regiomontanus  (or  Johann  Muller,  1436-1476  A.D.) 
was  a  German  and   studied  at   the    University  of    Vienna. 
After  doing  important  work  in  Germany  he  was  called  to 
Rome  by  the  Pope  to  reform  the  calendar  and  was  assas- 
sinated while  in  that  city.     The  ephemerides  calculated  by 


*  Ptolemy  entitled  his  work  fteyivrr)  ^a^art/cT;  <rvvTd£is,  or  "Greatest  Mathe- 
matical Collection."  The  book  was  translated  by  the  Arabs  into  their  language  and 
used  by  them  as  a  text-book.  The  name  Almagest  comes  from  a  blending  of  the 
Arabic  article  "  al  "  (the)  with  the  Greek  word  ^^lar-r\  (greatest). 


164  TRIGONOMETRY 

Regiomontanus  were  used  by  Columbus  in  crossing  the 
Atlantic.  Regiomontanus  wrote  a  text-book  entitled  De 
TrianguliSy  in  which  he  freed  the  subject  of  trigonometry 
from  its  astronomical  bondage.  Though  he  made  trigonom- 
etry a  part  of  geometry,  he  presented  the  subject  essentially 
in  the  form  in  which  it  is  customary  even  yet  to  make  a  first 
presentation  of  it  to  pupils. 

Several  other  Germans,  as  Pitiscus,  Rheticus,  and  several 
French  and  English  mathematicians  made  important  con- 
tributions to  the  development  of  trigonometry,  but  the 
thinker  who  first  put  the  subject  on  a  firm  modern  basis  was 

123.  Euler    (1707-1783),  born    in    Basle,    Switzerland. 
Euler's  life  as  a  scientific  worker  was  spent  mainly  at  St. 
Petersburg  and  Berlin.     Through  his  writings  and  influence 
trigonometry  was  established  as  an  independent  science. 

Since  Euler,  a  large  number  of  mathematicians  have  made 
contributions  to  trigonometry  in  the  larger  sense,  that  is, 
considered  as  a  branch  of  the  theory  of  functions,  which  has 
been  mentioned  merely  in  an  incidental  way  in  this  book. 

HISTORY  OF  TRIGONOMETRICAL,  FUNCTIONS 
AND  THEIR   NOTATION 

124.  Sine.    During  all  the  early  history  of  trigonometry,  the 
trigonometric  functions  were  regarded  as  lines,  not  as  ratios. 

v  Hipparchus  (120  B.C.)  used  but  one  trigono- 

metric  function.      This  was  the  chord  subtended 
A  by  double  the  angle,  and  it  therefore  corresponded 
in  a  general  way  to  the  sine  of  an  angle.     Thus, 

the  angle  AOP  was  regarded  as  determined  by 
FIG.  93.       the  chord  PQ 

Ptolemy  (150  A.D.)  treated  angles  by  the  same  method  as 
Hipparchus,  that  is,  by  use  of  the  chord  of  the  double  angle. 

This  method  introduced  unnecessary  labor  in  two  ways : 
first,  it  made  it  necessary  to  double  each  angle  dealt  with,  in 


HISTORY  OF   TRIGONOMETRY  165 

order  to  get  the  required  chord ;  second,  it  made  it  necessary 
to  divide  by  two  each  angle  obtained  as  the  result  of  a  process. 

The  Hindoos  regarded  an  angle  as  determined  by  the  semi- 
chord  of  twice  the  angle;  thus  by  them  in  the  above  figure 
the  angle  AOP  would  be  regarded  as  determined  by  PR. 
This  is  the  method  which  is  used  at  present  when  the  sine 
is  regarded  as  a  line. 

The  Arabs  also  determined  the  angle  by  the  semichord  of 
twice  the  angle,  one  of  their  writers  remarking  that  the  use 
of  the  semichord  "  saves  the  continual  doubling  "  mentioned 
above. 

Rheticus  (Germany,  1514-1576)  was  the  first  to  consider 
the  right  triangle  OPE  as  independent  of  any  arc  or  circle. 
He  defined  the  trigonometric  functions  as  ratios  of  the 
sides  of  the  right  triangle,  but  this  improvement  was  not 
adopted  by  other  mathematicians  until  the  time  of  Euler. 

Euler  also  defined  the  sine  and  other  trigonometric 
functions  as  ratios  between  the  sides  of  a  right  triangle. 
He  was  thus  able  to  make  them  functions  of  the  angle  only 
and  to  treat  them  as  pure  numbers.  In  this  way,  trigonom- 
etry became  an  independent  science. 

125.  Other  Functions.  The  Egyptians  used  the  cosine  and 
cotangent,  in  effect. 

Hero,  of  Alexandria  (110  B.C.),  in  effect,  used  a  table  of 
cotangents  by  which  to  determine  the  areas  of  regular 
polygons. 

The  Hindoos  used  the  versed  sine  and  cosine  as  well  as  the 
sine. 

The  Arabs  invented  the  tangent,  cotangent,  and  secant, 
though  these  functions  were  afterward  neglected  and 
reinvented  in  Europe. 

Regiomontanus  rediscovered  the  tangent  and  cotangent. 

Rheticu?,  using  the  simply  right  triangle,  had  the  secant 
and  cosecant  suggested  to  him  by  it. 


166  TRIGONOMETRY 

126.    Notation    of    the    Trigonometric    Functions.       The 

Egyptians  used  the  word  segt  for  both  the  ratios  employed 
by  them  (cosine  and  tangent). 

The  Hindoos  called  the  chord  jiva;  the  semi-chord,  or 
sine,  ardhajya,  and  later,  jiva  also ;  the  cosine  they  termed 
katijya,  and  the  versed  sine  utkramajya- 

The  Hindoo  word  for  sine,  jiva,  the  Arabs  transliterated 
as  jiba,  which  resembled  an  Arabic  word,  jaib,  meaning  an 
indentation  or  gulf.  The  Arabs  in  time  substituted  the 
latter  familiar  word  for  the  former  artificial  one.  Hence, 
when  the  Arabic  mathematical  works  were  translated  into 
Latin,  the  term  jaib  was  designated  by  the  Latin  word  sinus 
(which  means  "gulf"). 

Later,  Rheticus,  in  his  use  of  the  right  triangle,  termed 
the  sine  the  perpendicular,  and  the  cosine  the  basis. 

By  others  the  cosine  was  sometimes  termed  the  sinus 
rectus  secundus,  and  sometimes  the  complementi  sinus. 

Gunter  (England,  1580-1626)  was  the  first  to  use  the  word 
cosine,  which  he  obtained  by  contracting  the  words  "  comple- 
menti sinus." 

The  Arabs  called  the  tangent  umbra,  and  the  secant, 
diameter  umbrae,  as  a  result  of  their  use  of  these  functions  in 
connection  with  the  shadows  of  tall  objects. 

Later  in  Europe  the  tangent  was  sometimes  spoken  of  as 
the  umbra  recta,  and  the  cotangent  as  the  umbra  versa. 

The  words  tangent  and  secant  for  the  corresponding  trigo- 
nometric functions  were  first  used  by  Thomas  Finck  (Den- 
mark, 1583). 

Gunter,  who  invented  the  word  cosine,  also  invented  the 
word  cotangent. 

Girard  (Holland,  1590-1633)  was  the  first  to  use  the  ab- 
breviations sin,  tan,  sec,  etc.  These  abbreviations,  however, 
were  not  generally  accepted  till  they  were  taken  up  (1748) 
by  Simpson  in  England  and  Euler  in  Germany. 


HISTORY   OF   TRIGONOMETRY  167 

HISTORY   OP   TRIGONOMETRICAL   TABLES 

127.  History    of    Methods    of    Measuring    Angles.       The 

division  of  the  circumference  of  a  circle  into  360  degrees, 
each  degree  into  60  minutes,  and  each  minute  into  60  sec- 
onds, is  due  to  the  Babylonians.  This  system  of  angular 
measurement  was  transmitted  from  the  Babylonians  to  the 
Greeks,  Hindoos,  and  Arabs.  The  terms  minutes  and  seconds 
are  derived  from  ,  their  Latin  names  which  were  in  full 
"partes  minutse  primse"  and  "  partes  minutae  secundse." 

This  so-called  sexagesimal  notation  also  came  to  be  applied 
to  other  lines  and  quantities  than  the  circumference  of  a 
circle  as  we  shall  see  later. 

The  Hindoos  developed  the  Babylonian  sexagesimal  method 
into  a  rude  form  of  the  circular  method  of  measuring  angles 
(see  Art.  128).  The  circular  method  in  its  present  form  (use 
of  radians,  etc.)  came  into  use  in  the  early  part  of  the 
eighteenth  century. 

The  inventors  of  the  metric  system  of  weights  and  measures  at  the 
time  of  the  French  Revolution  proposed  to  divide  the  right  angle  into 
100  equal  parts  called  "grades,"  and  to  subdivide  the  grade  decimally, 
but  this  system  never  came  into  practical  use.  At  present  the  custom 
of  dividing  a  right  angle  into  90  degrees,  and  then  dividing  each  degree 
decimally  (instead  of  into  minutes  and  seconds),  is  growing  in  favor. 

128.  Notation^  used  in  Trigonometric  Tables.     As  decimal 
fractions  in  their  present  form  are  a  comparatively  modern 
invention,  in  the  early  history  of  Trigonometry  the  values 
of  the  trigonometrical  functions  were  necessarily  expressed 
in  some  other  way.     Thus  the  Greeks  used  sexagesimal  frac- 
tions in  expressing  the  lengths  of  the  lines  which  were  their 
trigonometrical  functions.     Ptolemy  divided  the  diameter  of 
the  circle  into  120  equal  parts,  each  of  these  parts  into  60 
minutes,  and  each  minute  into  60  seconds. 

For  instance,  where  we  would  write  sine  18°  =  .3090,  Ptolemy  wrote 
chord  36°  =  37°  4' 55". 

The  Hindoos  divided  the  radius  of  the  circle  into  3148 


168  TRIGONOMETRY 

equal  parts,  3148  being  the  number  of  minutes  in  an  arc 
equal  to  the  radius.  Hence  the  Hindoos  made  an  approach 
to  the  circular  measure  for  angles,  the  number  denoting  the 
radius,  however,  in  their  use  of  the  relations  being  deter- 
mined by  the  angle  rather  than  the  unit  angle  by  the  radius. 
Regiomontanus  in  forming  his  tables  first  used  a  radius  of 
600,000,  but  later  he  used  a  purely  decimal  scale,  10,000,000 
being  the  radius.  Hence  his  work  may  be  regarded  as  a 
transition  from  the  sexagesimal  to  the  decimal  scale. 

129.  Computation  of  Trigonometrical  Tables.  Hipparchus 
(120  B.C.)  computed  a  table  of  chords  for  different  angles. 
This  table,  however,  has  been  lost. 

Ptolemy  in  his  Almagest  gives  a  table  of  chords  (computed 
in  sexagesimal  fractions  carried  out  to  a  point  equivalent  to 
5  decimal  places)  for  every  |°  of  the  quadrant,  the  table 
being  remarkably  accurate. 

Hero  of  Alexandria  (110  B.C.)  gives  a  table  of  cotangents 

calculated  for  cot  (-  -j  when  n=  3,  4,  .  .  .  12. 

The  Hindoos  (530  A.D.)  computed  a  table  of  sines  for 
every  3|°  of  the  quadrant. 

The  Arabs  (Bagdad,  980  A.D.)  formed  a  table  of  sines  for 
every  ^°,  and  also  a  table  of  tangents  and  cotangents. 

The  printing  press  was  invented  about  the  year  1450. 
Shortly  afterward  the  Germans  took  up  the  problem  of  com- 
puting very  full  and  exact  trigonometric  tables,  and  to  their 
industry  we  owe  our  tables  essentially  in  their  present  form. 

Peuerbach  (1423-1461),  teacher  of  Regiomontanus,  com- 
puted a  table  of  sines  for  every  10'  with  600,000  as  a  radius 
(i.e.  six-place  tables). 

Regiomontanus  constructed  a  table  of  sines  with  6,000,000 
and  another  with  10,000,000  as  the  radius. 

Regiomontanus  also  constructed  a  table  of  tangents  for 
every  1'  with  100,000  as  a  radius. 


HISTORY   OF   TRIGONOMETRY  169 

Apian  (1495-1552)  made  a  table  of  sines  for  every  1' 
with  a  radius  equal  to  100,000. 

Rheticus  computed  tables  of  sines,  tangents,  and  secants 
for  every  10"  with  radius  equal  to  10,000,000,000  ;  and  later 
a  table  of  sines  with  radius  equal  to  1015.  He  began  tables  of 
tangents  and  secants  on  the  same  scale,  but  died  before  com- 
pleting them.  In  this  work  he  employed  several  computers 
for  twelve  years  and  spent  large  sums  of  money.  When 
completed  by  his  pupil,  Otho,  and  published,  these  tables 
made  a  volume  of  1468  pages. 

Pitiscus  (1561-1613)  computed  tables  of  sines,  tangents, 
secants,  cosines,  cotangents,  cosecants,  with  radius  equal  to 
1025.  By  annexing  tables  of  proportional  parts,  he  facili- 
tated interpolations. 

It  is  to  be  remembered  that  each  time  we  use  trigonometric  tables 
we  use  again  the  labor  of  these  indefatigable  workers;  or,  to  put  it 
another  way.  by  a  species  of  kindly  foresight  on  the  part  of  these  men 
we  find  a  large  part  of  our  work  already  done  for  us  by  them. 

Lord  Napier  of  Scotland  published  his  invention  of 
logarithms  in  1614.  Immediately  upon  this  invention, 
logarithmic  tables  of  sines,  cosines,  tangents,  and  cotangents 
were  formed.  These  tables  were  printed  in  1633. 

130.    Methods  of  Computing  Trigonometric  Tables.     Hip- 

parchus  and  Ptolemy  in  construct- 
ing their  tables  of  chords  used  the 
theorem  of  geometry  which  reads  "  If 
a  quadrilateral  be  inscribed  in  a  circle, 
the  product  of  the  diagonals  equals 
the  sum  of  the  products  of  the  oppo- 
site sides ; "  i.e.  (Fig.  94)  AC*BD  = 
BCxAD  +  CDxAB.  By  means  of 
this  theorem,  if  the  chords  of  two 

arcs  are  known  (as  of  45°,  30°),  the  chords  of  the  sum  and  of 
the  difference  of  those  arcs  (i.e.  of  75°  and  15°)  can  be  com- 


170  TRIGONOMETRY 

puted.  Hence  the  theorem  in  a  rough  way  is  equivalent  to 
the  trigonometrical  formulas  for  sin  (A  ±  B)  and  cos  (A  ±  B) 
(Art.  71).  The  theorem  was  also  applied  by  Ptolemy  to  the 
problem  of  finding  the  chord  of  half  an  arc  when  the  chord 
of  the  whole  arc  was  known. 

Both  the  Hindoos  and  Germans  in  computing  their  tables 
of  trigonometric  functions  used  methods  which  were  essen- 
tially the  same  as  those  given  in  Art.  116.  As  has  been 
said,  much  more  expeditious  methods  are  now  at  the  service 
of  the  computer,  and  these  methods  have  been  used  in  veri- 
fying and  correcting  the  tables  as  at  first  computed. 

SOLUTION   OP   TRIANGLES 

131.  Greeks  (see  Ptolemy's  Almagest,  Book  1)  made  spher- 
ical trigonometry  primary  and  fundamental.  Plane  trigo- 
nometry was  developed  only  as  a  part  or  detail  of  spherical 
trigonometry.  The  methods  of  solving  spherical  triangles 
used  by  the  Greeks  were  mainly  geometrical  and  compara- 
tively awkward.  These  methods  are  derived  from  the 
principles  of  projection,  and  when  applied  to  right  spherical 
triangles  become  equivalent  to  four  of  the  ten  formulas 
which  are  included  in  Napier's  Rule  for  Circular  Parts. 

In  plane  trigonometry,  as  treated  by  the  Greeks,  a  right 
triangle  was  solved  by  inscribing  the  triangle  in  a  circle. 
An  oblique  triangle  was  solved  by  resolving  it  into  right  tri- 
angles. The  fundamental  principle  in  the  solution  of  plane 
oblique  triangles,  viz.  that  the  sides  are  to  each  other  as 
the  double  chords  of  double  the  angles  opposite  (i.e.  as  sines 
of  angles  opposite)  was  used  implicitly  by  Ptolemy,  but  was 
not  stated  by  him  in  so  many  words.  In  one  of  the  ex- 
amples solved  in  the  Almagest,  three  sides  of  an  oblique  tri- 
angle are  given,  and  the  triangle  is  solved  by  finding  the 
segments  of  one  of  the  sides  made  by  a  perpendicular  on  it 
from  the  opposite  vertex. 


HISTORY  OF   TRIGONOMETRY  171 

To  show  how  spherical  trigonometry  led  the  Greeks  to  plane  trigo- 
nometry, we  may  mention  one  of  the  problems  occurring  in  their  treat- 
ment of  the  former  subject,  viz:  To  divide  a  given  arc  into  two  parts 
so  that  the  chords  of  the  doubles  of  those  arcs  shall  have  a  given  ratio. 

Stated  in  terms  of  modern  notation  this  problem  is,  Given  x+y  = 

a  given  angle  (/),  to  find  x  and  y  so  that  -  -  =  -.    Stated  with  refer- 

siuy      b 

ence  to  the  triangle  ABC,  this  problem  becomes  one  in  Case  II  of 
oblique  plane  triangles;  for  /.  C  =180°-  (x  +  y)  =  180°  —j,  Z.A  =  x, 
a  AC=b. 


The  Hindoos,  like  the  Greeks,  made  use  of  trigonometry 
only  as  an  aid  in  the  study  of  astronomy.  They  solved  both 
plane  and  spherical  triangles,  but  treated  plane  trigonometry 
as  a  mere  detail  of  spherical  trigonometry. 

132.  The  Arabs  also  gave  spherical  trigonometry  the  lead- 
ing  place    in   the    study   of  the    subject.     They   simplified 
Ptolemy's  method  of  solving  spherical  triangles,  discovered 

that   in  spherical    triangles  cos  A  =  °QS  a  ~  COS  b  COS  C,  and  to 

sm  6  sin  c 

the  four  of  the  ten  formulas  included  in  Napier's  Rule  for 
Circular  Parts,  which  Ptolemy  had  implicitly  known,  added 
two  others,  viz.  : 

cos  B  =  cos  b  sin  A,     cos  c  =  cot  A  cot  B. 
The  Arabs,  however,  developed  no  general  theory  for  the 
solution  of  plane  or  spherical  triangles. 

133.  Regiomontanus  separated  plane  from  spherical  trigo- 
nometry  and    made   plane   trigonometry   primary.     In  his 
treatise  he  begins  with  the  right  triangle,  solves  it  by  using 
the   sine  function  only,  and  then  solves  equilateral  and  isos- 
celes triangles  by  resolving  them  into   right  triangles.     He 
also  solves  oblique   triangles   much  as  is   done    at  present. 
His  treatment  of  spherical  trigonometry,  however,  is  far  less 
general  and  satisfactory. 

Romanus  (Belgium,  1561-1625)  condensed  the  twenty-six 
cases  of  spherical  trigonometry  then  in  use  into  six  cases. 


172  TRIGONOMETRY 

134.  Lord  Napier  (Scotland,  1550-1617)  reduced  the  solu- 
tion of  right  spherical  triangles  to  ideal  simplicity  by  his 
Rule    for    Circular  Parts.     This    has    been    commended    as 
perhaps  "  the  happiest  example  of  artificial  memory  that  is 
known."     He  also  simplified  the  solution  of  oblique  spherical 
triangles  by  his  discovery  of  the  formulas  known  as  Napier's 
Analogies. 

135.  Notation  of  Triangles.     To  Euler  is  due  the  method 
of  denoting   the  angles  of  a  triangle  by  the  capital  letters 
Ay  B,  C,  and  the  sides  opposite  by  the  small  letters  a,  6,  c. 

136.  The  theory  of  the  complete  spherical  triangle,  that 
is,  of  the  triangle  in  which  the  length  of  the  sides  is  not  nec- 
essarily less  than  180°,  was  developed  by  Gauss  (Germany, 
1777-1855)  and  Moebius  (Germany,  1790-1868),  but  such 
triangles  are  not  much  used  in  practice. 

137.  Spheroidal   trigonometry,  that  is,  the  theory  of  tri- 
angles  on    the   surface    of   a    spheroid   has    great  practical 
importance  because  of  its  use  in  surveying  large  portions  of 
the  earth's  surface,  as  in  the  coast  and  geodetic  surveys  in 
different  countries. 

DEVELOPMENT   OF    GONIOMETRY 

138.  Greeks.     As  has  been  stated  (Art.  130),  the  geomet- 
rical methods  used  by  the  Greeks  in  constructing  tables  of 
chords  were    in    a  rough  way  equivalent  to  a  use  of  the 
formulas  for  sin  (A±B),  cos  (A±J5),  and  sinj  A- 

139.  The  Hindoos  knew  the  identical  equation 


sin2 


They  also  used  the  formula  sin  \A  =  Vl719(3438-cos  A), 
where  3438'  is  the  radius  of  the  circle.     This  is  equivalent  to 


the  formula  sin     A 


jl  —  cos  A 
-  \ 9 ' 


HISTORY  OF   TRIGONOMETRY  173 

In  computing    trigonometric    tables  they  appear  to  have 
used  the  formula 

sin  (n  +  1)  a  —  sin  na  =  sin  na  —  sin  (?i  —  1)  a  —  sin  na  cosec  a. 

This   formula  is   not   quite  accurate   and    was   probably 
arrived  at  inductively. 

140.    The  Arabs  knew  the  relations 

.      sin  6       ,   ,      cos  </> 

tan  &  =  -    -s  cot  o>  =  —    -£ 

cos  <f)  sin  0 

and  were   also    able    to    solve    an    equation    like    tan  </>  =  a, 
obtaining  sin  </>  = 


141.  Rheticus   obtained  the  formulas 

sin  2  A  =  2  sin  A  cos  A9 
sin  3  A  =  3  sin  A  —  4  sin8A 

Romanus  discovered  the  formula  for  sin  (A  +  B). 

The  formulas  for  sin  (A  —  B)  and  cos  (A±B)  were  published 
byPitiscus(1599). 

142.  Vieta  (France,  1540-1603)  gave  the  general  formulas 
for  sin  nA  and  cos  nA  in  terms  of  sin  A  and  cos  A- 

OTHER  PROCESSES 

143.  Trigonometrical  Series-     The  series  for  sin  x  and  cos  x 
in  terms  of  powers  of  x  and  for  sin"1  x  in  terms  of  sin  x  were 
known  to  Sir  Isaac  Newton  before  the  year  1669. 

Those  for  tan  x  and  sec  x  in  terms  of  powers  of  x  and  for 
tan"1  x  in  terms  of  powers  of  tan  x  were  discovered  by 
Gregory  (England,  1638-1675)  in  1670.  - 

144.  Inverse    Circular    Functions    in    their    general    form 
were  introduced  by  John   Bernouilli  (1667-1748). 


174  TRIGONOMETRY 

145-  Use  of  V  — 1  or  i.  John  Bernoulli!  first  treated 
trigonometry  as  a  branch  of  analysis.  Among  other  alge- 
braic methods  he  introduced  the  use  of  V— ly  or  i9  into 
trigonometry  and  obtained  real  results  by  its  use.  For 
instance,  by  employing  V—  1  he  obtained  a  series  for  tann<£ 
in  term  of  powers  of  tan  (f>. 

This  use  of  i  was  followed  up  by  Euler,  who  among  other 
results  obtained  the  formula 

(sin  x  +  i  cos  x)n  =  sin  nx  4-  i  cos  nx 
known  as  De  Moivre's  Theorem. 


EXERCISE  48.    GENERAL   REVIEW 

1.  Simplify  Iog2  4  +  5  Iog3  9  +  i  logic  -1  -  logM ViOOl. 

2.  Compute  the  value  of  x  from  the  equation  5  x3  =  •\/'.2784 

3.  Also  from  cos  x  =  (.9387)*. 

(7.605)8VlO2 

4.  Also  from  tan  x  =  ^—  -  • 

(27.32)* 

5.  If  x  is  an  angle  in  the  first  quadrant  and  cos#  =  T87,  find  the 

value  of  sinx  +  tana:. 
cos  x  —  cot  x 

6.  If  x  is  an  angle  in  the  first  quadrant  arid  2  cos  x  =  2  —  sin  x,  find 
the  value  of  tan  x. 

7.  If  tan  x  =  -,  find  sin  2  cc. 

6 

8.  If  sin  y  =  a  and  tan  y  =  b,  prove  that  (1  —  a2)  (1  +  62)  =  1. 

9.  ABCD  is  a  square.     D  is  joined  to  _EJ,  the  midpoint  of  AB.    Find 
the  trigonometric  ratios  of  Z  ECD. 

10.  Determine  the  numerical  value  of  sin  18°  by  use  of  the  geometric 
method  of  inscribing  a  regular  decagon  in  a  circle. 

11.  If  A  is  an  angle  in  the  first  quadrant  and  tan  A  =  £,  find  the 

value  of  J"*»^-9sin^ 
p  cos  A  -f-  g  sin  ^4 

12.  Which  of  the  following  statements  are  possible  and  which  im- 
possible : 

(1)  16  sin  a;  =1.  (2)  4  sec  0  =  1.  (3)  7  tan  #  =  30. 


GENERAL   REVIEW   EXERCISE  175 

13.  Prove  that  sec  x  +  tan  x  =  sec2  x  +  sec  x  tan  x  +  tan  x. 

tan  a;  +  sec  x 

14.  Prove  that  ™™*x  =    2  sm  x    —  sin  a;. 

sin  x       1  -j-  cos  x 

15.  Find  the  numerical  value  of  3  tan3  30°  sec3  60°  sin2  90°  tan2  45°  + 
5  cos  90°. 

16.  If  tan2  45°  -  cos2  60°  =  y  sin  45°  cos  45°  tan  60°,  find  y. 

cos2  -  sec  ^  tan  - 

17.  If  a;  sin -cos8- =  -  J,  find  as, 

csc  4cose 


Solve  each  of  the  following  right  triangles,  given  : 

18.  A  =  36°  18'  6"  [36.3°],  b  =  217.9  ft. 

19.  6  =  315.92  ft.,c  =  814.23  ft.    21.   B  =  12°  15'  [12.25°],  c==  1001.4. 

20.  c  =  900,  b  =  887.  22.   ^4  =  1°  20'  [1.33°],  c  =  872.56. 

23.  In  a  right  triangle  b  =  426,  J.  =  38°  45'  [38.75°].     Find  a  -f  c  and 
the  area. 

24.  The  hypotenuse  of  a  right  triangle  is  5  ft.  and  one  angle  of  the 
triangle  is  30°.     Solve  the  triangle  and  find  the  area  without  the  use  of 
tables. 

25.  The  area  of  a  regular  polygon  of  11  sides  is  80.     Find  the  side, 
radius,  and  apotheni  of  the  polygon. 

26.  In  an  isosceles  triangle  the  leg  is  21.7  and  the  area  32.51. 
Solve  the  triangle. 

27.  The  legs  of  a  right  triangle  are  to  each  other  as  5  :  9.     Find  the 
angles  of  the  triangle. 

28.  On  the  steepest  part  of  the  Mt.  Washington  railway  (Jacob's 
Ladder),  there  is  a  rise  of  13^  inches  for  every  3  ft.  of  track.     What 
angle  does  the  track  make  with  the  horizontal?     At  this  rate  what 
would  be  the  rise  in  one  mile  of  track? 

Show  that  in  a  right  triangle  : 

29.  cos2^  =  ^^.  30.    sm3A- 


c3 
31.     sm^-sin.B2+c 


176  TRIGONOMETRY 

32.  Find  the  other  trigonometric  functions  of  A,  when  cos  A  =  —  -| 
and  A  lies  between  540°  and  630°. 

33.  Given  sec  x  =  —  f  and  x  in  the  third  quadrant,  find  the  value  of 
sin  x  +  tan  x 

^os  x  +  cot  x 

34.  Find  the  trigonometric  functions  of  180°  +  x  and  of  270°  —  x 
when  tan  #  —  \. 

35.  For  what  values  of  x  between  0°  and  360°  is  sin  x-{-  cos  x  positive, 
and  for  what  values  is  it  negative? 

36.  Find  the  numerical  value  of 

3  sin2  225°  +  4  sin  (-  120°)  tan  150°  -  \  cos2 330°  cot  750°  +  5  sin2 180°. 

37.  For  each  of  the  following  angles  state  which  of  the  three  princi- 
pal trigonometric  ratios  are  positive  : 

(1)  460°.  (2)  -220°.  (3)   -1200°.  (4)   ^  • 

38.  Trace  the  changes  in  sign  and  magnitude  of 

sin  A  between  0°  and  360°. 

esc  A  between  0  and  IT. 

cos  x  between  ?r  and  2  ?r. 

tan  A  between  -  90°  and  -  270°. 

39.  If  A  is  in  the  third  quadrant  and  tan  A  =  -f^,  find  the  value  of 
sin  2  A 

40.  Express  the  cosine  of  an  angle  in  the  second  quadrant  in  terms 
of  (a)  each  of  the  other  trigonometric  functions  of  the  given  angle, 
(b)  the  cosine  of  the  complement  of  the  angle. 


41.  If  sin  A  =  -}-|  and  sin  B  =  f  ,  and  A  and  B  are  both  acute,  find 
the  numerical  value  of  tan  (A  +  B)\  also  of  tan  (A  —  B). 

42.  If  x  is  an  angle  in  the  second  quadrant  and  sin  x  =  f  ,  find  the 
value  of  sin  2  x  +  cos  2  x. 

2  Q         5  B 

43.  Express  2  cos  —  cos  —  as  a  sum  or  difference. 

o  o 

44.  If  sin  \  x  =  1,  find  the  numerical  value  of  cos  x.     Also  of  tan  x. 

Prove  that: 

45.  sin2  (A  +  B)  —  sin2  (A  —  B)  =  sin  2  A  sin  2  B. 


46.  =  cot$a;.  47.    sin  50°  +  sin  10°  =  sin  70°. 

cos  3  x  —  cos  4  a? 


GENERAL   REVIEW  EXERCISE  177 

48.    sin2 15°  + cos2 15°  =  1.  49.    cos  55°  +  sin  25°  =  sin  85°. 

50. 

cos  A  +  cos  2  A  -f-  cos  3  A 

51'    l  +  tan2(45°-x)  = 

0\_cosf--f 


52. 


smi 


Solve  each  of  the  following  oblique  triangles,  given : 

53.  A  =  30°  18'  12"  [39.3°],  b  =  3294,  c  =  2846. 

54.  .4  =  76°  24'  36"  [76.41°],  B  =  48°  42'  [48.7°],  c  =  1012. 

55.  a  =  850,  b  =  760,  c  =  590. 

56.  B  =  46°  18'  [46.3°],  b  =  213.76,  a  =  192.72. 

57.  b  =  927,  ^4  =  79°,  B  =  21°  17'  12"  [21.29°]. 

58.  a  =  V3,  6  =  V2,  c  =  V5. 

59.  ^  =  51°  30'  [51.5°],  a  =  294.6,  6  =  301.7. 

60.  a  =  926.8,  6  =  842.5,  C=  46°  27'  [46.45°]. 

61.  Solve  the  triangle  in  which  K=  20.602,  a  =  214.2,  and  b  =  315.8. 

62.  The  diagonals  of  a  parallelogram  are  347  and  264  ft.,  and  the 
area  of  the  parallelogram  is  40.437  sq.  ft.     Find  the  sides  and  angles 
of  the  parallelogram. 

63.  The  diagonals  of  a  quadrilateral  are  34  and  56,  and  they  inter- 
sect at  an  angle  of  67°.     Find  the  area  of  the  quadrilateral. 


Solve  the  following  equations  for  answers  not  greater  than  360°  or 
less  than  0° : 

64.  sec  x  -f  tan  x  =  ±  V3.  67.   2  sin  x  sin  3  x  —  sin2  2  x  =  0. 

65.  sec2  x  +  cot2  x  =  */.  68.    sin  2  0  +  sin  0  =  cos  2  6  +  cos  0. 

66.  sin  2  x  —  V3  cos  x.  69.    sin  2  y  +  V3cos  2  y  =  1. 

70.  sin(600-z)-sin(60°+x)  =  iV3. 

71.  Give  the  answers  to  Exs.  64-70,  in  the  unlimited  form. 


178  TRIGONOMETRY 

72.  If  2  cos2  x  —  1  cos  x  +  3  =  0,  show  that  there  is  only  one  value  for 
cos  a;. 

73.  Find  the  least  possible  positive  value  of  0  which  will  satisfy  the 
equation  2  V3  cos2  6  =  sin  6. 

74.  Solve  sin  x  •+•  sin  2  x  +  sin  3  x  =  1  4-  cos  x  +  cos  2  a?. 

75.  If  sin  3  a;  +  sin  2  a?  =  sin  x,  find  tan  #. 


76.  Find  the  length  of  an  arc  intercepted  by  an  angle  of  2.2  radians 
at  the  center  of  a  circle  whose  radius  is  5  ft.     How  many  degrees  in 
this  angle  ? 

77.  Two  angles  of  a  triangle  are  .5  and  .4  radians.     Find  the  third 
angle  in  radians  and  in  degrees. 

78.  The  sura  of   two   angles  is  2   radians,   their   difference  is  10°. 
How  many  radians  are  there  in  each  of  these  angles  ? 

79.  Prove  cos  f—  +  x\- cos  f—  -  x\  =  2  sin  x. 

Q  K  "1  Q 

80.  Find  the  numerical  value  of  -  sin2  -  +  4  cos2  —  -  ^  tan2  —  - 


81.  If  sin /x  +  ^  jsinfa  — ^ )  =  -,  find  #. 

N  /  ./ 

82.  Simplify  tan  (—  -  x]  +  tan ( —  +  A 

\4          )  \ 4         ) 

83.  An  angle  of  30°  at  the  center  of  a  circle  subtends  an  arc  AB  of 

length  -  ft.     Find  the  length  of  the  perpendicular  dropped  from  A  on 
3 


84.  Express  each  of  the  following  angles  in  degrees  : 
sin-1!-;  COS-4V2;  tan-^-l);  sin^-l);  coa- 

85.  Find  tan 


86.  Prove  that  tan-1  2  +  tan-1!  =  |  • 

87.  Find  the  value  of  x,  if  tan-1  x  +  2  cot"1  a;  =  ~  • 

o 

88.  How  many  degrees  in  sin-1(—  |-V2)  ?     How  many  radians  ? 

89.  Prove  sin"1  a  =  sec"1  —  • 

Vl-a2 


GENERAL   REVIEW   EXERCISE  179 

90.    Solve  the  following  for  x  and  y : 

sin-1  x  +  sin-1  y  =  120°.  cos-1  x  -  cos-1  y  =  60°. 


91.  At  a  point  50  ft.  from  the  base  of  a  tower  the  angle  of  eleva- 
tion of  the  top  of  the  tower  was  found  by  the  use  of  a  transit  instru- 
ment to  be  68°  18'  [68.3°].     If  the  height  of  the  instrument  above  the 
ground  was  4.75  ft.,  what  was  the  height  of  the  tower  ? 

92.  If  the  railway  up  Pike's  Peak  rises  7552  ft.  in  8J  mi.,  what 
angle  does  the  railway  make  with  the  horizon  on  the  average  ? 

93.  Two  towers  are  240  and  80  ft.  high,  respectively.     From  the 
foot  of  the  second  the  angle  of  elevation  of  the  top  of  the  first  is  60°. 
Find,  without  the  use  of  tables,  the  angle  of  elevation  of  the  second 
from  the  foot  of  the  first. 

94.  An  unknown  force  combined  with  one  of  128  Ib.  produces  a 
resultant  force  of  200  Ib.     The  resultant  makes  an  angle  of   18°  24' 
[18.4°]  with  the  known  force.     Find  the  magnitude  of  the  unknown 
force  and  the  angle  which  it  makes  with  the  known  force. 

95.  A  tree  82  ft.  high  stands  at  one  corner  of  a  garden  which  is  in 
the  form  of  an  equilateral  triangle.     The  distance  from  the  top  of  the 
tree  to  the  midpoint  of  the  opposite  side  of  the  garden  is  112  ft.     Find 
a  side  of  the  garden. 

96.  If  the  earth's  radius  (3956  mi.)  as  viewed  from  the  sun  sub- 
tends an  angle  of  8.8",  find  the  distance  of  the  earth  from  the  sun. 

97.  In  a  circle  whose  radius  is  13.7,  find  the  area  of  a  segment 
whose  angle  is  —  -  radians. 

98.  In  order  to  determine  the  breadth  of  a  river,  a  base  line  of  500 
yd.  was  measured  on  one  shore,  and  at  each  end  of  the  base  line  the  angle 
included  between  the  base  line  and  a  line  to  a  rod  on  the  other  bank 
was  measured.     These  angles  were  found  to  be  53°  and  79°  12'  [79.2°], 
respectively.     What  was  the  breadth  of  the  river  ? 

99.  If  a  barn  is  40  X  80  ft.,  and  the  pitch  of  the  roof  is  45°,  find 
the  length  of  the  rafters  and  the  area  of  the  entire  roof,  the  horizon- 
tal projection  of  the  cornice  being  1  ft. 

100.  If  the  sun's  angle  of  elevation  is  60°,  what  angle  must  a  stick 
make  with  the  horizontal  in  order  that  its  shadow  on  a  horizontal 
plane  may  be  the  largest  possible. 


180  TRIGONOMETRY 

101.  If  a  railroad  rises  1  ft.  for  every  1000  ft.  of  its  length,  what 
angle  does  it  make  with  the  horizontal  ? 

102.  In  surveying  a  circular   railroad   curve   successive  chords  of 
100  ft.  each  are  laid  off.     Find  the  radius  of  the  curve,  if  the  angle 
between  two  successive  chords  is  177°. 

103.  If  the  diagonal  of  a  regular  pentagon  is  32.835,  what  is  the 
radius  of  the  circumscribed  circle  ? 


104.  The  angle  x  is  in  the  third  quadrant  and  cos  x  =  —  f  ;  find  the 
value  of  esc  x,  tan  x,  sin  ^  x,  tan  (180°  —  x),  and  sin  —  x. 

105.  Find  all  the  values  of  x  between  0°  and  360°  which  satisfy  the 
equation  sin  (30°  -  a;)  =  cos  (30°  +  x). 

106.  If  x  is  an  angle  in  the  second  quadrant,  prove  geometrically 
that  tan  (270°  +  x)  =  —  cot  x. 

107.  One  angle  of  a  rhombus  is  60°  and  the  opposite  diagonal  is  5 
inches.     Without  the  use  of  tables  find  the  sides  of  the  rhombus  and 
its  area. 

108.  Give  a  general  formula  for  all  angles  whose  sine  is  -J-.     Is  —  J. 
Is  -1. 

109.  Express  cos  2x  in  terms  of  each  of  the  functions  of  x. 

110.  Express  cos  A  cos  B  as  a  sum. 

111.  If  cos  A  =  h,  and  tan  A  =  k,  find  the  equation  connecting  h  and  Jc. 

112.  How  many  radians  in  each  interior  angle  of  a  regular  hexagon  ? 
In  each  exterior  angle  ?     How  many  degrees  in  each  of  these  angles  ? 

113.  Prove  that  cos'1  f  f  +  .2  tan'1  i  =  sin-1  f  . 


114.  , 

3  an  x  —  cos  x 


115.  In  the  isosceles  right  triangle  ABC,  D  is  the  midpoint  of  AC. 
Prove  without  the  use  of  tables  that  cot  /.  ABD  :  cot  Z  DEC  =2:3. 

116.  If  6  lies  between  180°  and  270°,  and  3  tan  0  =  4,  find  the  value 
of  2  cot  0  =  —  5  cos  0  4-  sin  0. 

117.  Is  it  possible  to  have  an  angle  whose  tan  is  503  ?     Whose  cos 
is  |  ?     Whose  secant  is  ^  ?     Whose  sine  is  23  ? 

118.  Show  that  cos  80°  +  cos  40°  —  cos  20°  =  0. 


119.    That  2  sin  f  x  +  -    sin  (  x  —  -  ==  sin2  x  —  cos2  x. 


GENERAL  REVIEW   EXERCISE  181 

120.  If  sin  (60°  -  x)  -  sin  (60°  +  x)  ==  i  V3,  find  tan  2  a. 

121.  Express  2  sin  9  A  sin  ^4  in  the  form  of  a  sum  or  difference. 

122.  Find   the   value   of   sin^i  +  Stau^jVS  —  2cot-1l  +  sec-1l, 
using  values  between  0°  and  90°  • 

123.  If  tan  2x  =  -2T4-,  find  tan  x  and  sin  x,  it  being  given  that  x  is  an 
angle  in  the  third  quadrant. 

124.  Find  by  inspection  one  value  of  x  when 

cos  (10°  +  A]  cos  (10°  —  A)  +  sin  (10°  +  A)  sin  (10°  -A)  =  cos  x. 

125.  A  surveyor  standing  on  a  bank  of  a  river  observes  the  angle 
subtended  by  a  flagpole  on  the  opposite  bank  to  be  33°  10'  [33.17°] 
and  when  he  retires  120  ft.  from  the  bank  he  finds  the  angle  to  be 
18°  16'  [18.27°].     Find  the  width  of  the  river. 

126.  Develop  cos  (270°  —  x  —  y)  in  the  shortest  way. 

127.  What  is  the  angle  of  elevation  of  the  sun  when  the  length  of 
the  shadow  of  a  pole  is  V3  times  the  height  of  the  pole? 

128.  If  tan  A  =  f  and  sin  B  =  if,  and  A  is  in  the  third  quadrant 
and  B  in  the  second,  find  sin  (A  -f  JS),  cos  (A  +  .B),  tan  (A  -f-  -B). 

129.  At  the  Panama  Canal  the  Gatun  dam  has  three  different  slopes : 
the  ratio  of  the  horizontal  to  the  vertical  near  the  base  is  16  to  1 ;  in 
the  middle  of  the  dam  this  ratio  is  8  to  1 ;  and  at  the  top  the  ratio  is 
4 : 1.     What  three  different  angles  does  the  surface  of  the  dam  make 
with  the  horizontal  ? 

130.  If  A  is  an  angle  in  the  first  quadrant,  and  sec2  A  esc2  A  —  4  =  0, 
find  the  numerical  value  of  cot  A. 

131.  If  0  is  an  angle  in  the  third  quadrant,  and  sec2  0  =  2  +  2  tan  0, 
find  sin  0. 

132.  Find  all  the  values  of  x  between  0°  and  500°  which  satisfy  the 
equation  tan  (45°  —  x)  +  cot  (45°  —  x)  =  4. 

133.  Graph  y  =  sin"1  x.  134.    Also,  y  =  tan'1  x. 

135.  From. the  top  of  a' mountain  3  mi.  high,  the  angle  of  depression 
of  the  horizon  is  2°  13'  50"  [2.23°].     Hence  determine  the  diameter  of 
the  earth. 

136.  Can  an  angle  exist  such  that  9  sin  2  x  +  3  sin  x  =  20  ?     Why  ? 

137.  Find  the  numerical  value  of  tan2  —  -+-  cos2— -^  -f-  sin2^- 

3  46- 

138.  Find  the  sines  of  all  angles  less  than  2  TT  whose  tangents  are 
equal  to  cos  135°. 


182  TRIGONOMETRY 

139.  Given  cos  f  -  +  x  J  =  a,  find  cot  f—  +  x\  • 

140.  What  is  the  most  general  value  of  x  which  satisfies  both  of  the 
equations  cot  x  =  —  V3  and  esc  x  =  —  2. 

141.  Show  that  2  sin  f-  +  A\  cos  f-  -f  B\  =  cos  (  J.  +  B)  +  sin  (4  -  J3). 

142.  Find  the  length  of  a  circular  arc  whose  radius  is  5  ft.  and 
whose  subtending  angle  is  3  units  of  circular  measure. 

143.  In  the  triangle  ABC,  B  is  45°,  and  C  is  120°,  and  a  is  40. 
Without  the  use  of  tables  find  the  length  of  the  perpendicular  drawn 
from  A  to  BC  produced. 

144.  Prove  that  -T™i*±Si»**-  =  tan  a. 

1  -f  cos  x  -f-  cos  2  x 

145.  When  y  =  —  ^,  find  the  numerical  value  of 

4 

sin2  y  —  cos2  y  +  2  tan  y  —  sec2  y. 

146.  Prove  the  identity  sin"1  y  +  tan-1  y  =  sin' 


VT 

147.  Is  sin  a;  —  2  cos  x  +  3  sin  a?  —  6  =  0  a  possible  equation  ? 

148.  A  vertical  pole  stands  at  the  center  of  a  circular  mill  pond  and 
rises  100  ft.  above  the  surface  of  the  water.    From  a  point  on  the  shore 
the  angle  of  elevation  of  the  top  of  the  pole  is  20°.     Find  the  area  of 
the  pond. 

149.  When  the  planet  Venus  is  most  brilliant,  its  diameter  subtends 
an  angle  of  40"  as  seen  from  the  earth.     If  the  diameter  of  the  planet 
is  7600  mi.,  what  is  the  distance  of  the  planet  from  the  earth  at  such 
a  time? 

150.  Verify  the  statement 

-cot2-  +  3sin2?-2csc2^-?tan2^  =  —  • 
36  3  3463 

151.  Find  the  value  of  sin  x,  if  tan  (  -  +  «  ]  tan  (  ^  —  x  )  -f  2  =  0. 

\3       J        \3       / 

152.  What  sign  has  sin  x  cos  x  for  the  following  values  of  x  :  140% 
278°,  -356°,  -1125°? 

153.  If  1  +  sin2  x  =  3  sin  x  cos  x,  find  tan  x. 

1£4.    If  i  denotes  the  angle  of  incidence  of  a  ray  of  light  falling  on 

water,  and  r  the  angle  of  refraction,  and  ^-^  =  1.423,  find  r  when 
<  =  34.37°.  smr 


GENERAL  REVIEW   EXERCISE  183 

a2  -4-  b2 

155.  When  is  sin#  =  —    -  —  possible,  and  when  impossible? 

156.  Show  that 


157.  Solve  sin  2  x  —  cos  2  x  —  sin  x  -\-  cos  x  =  0. 

158.  Solve  x  =  sin"1  1  +  tan~J  1. 

159.  Trace  the  changes  in  sign  and  magnitude  of   -    -  -  as  x  in- 

cos  2  0 
creases  from  0  to  -  • 

160.  Two   trains   leave   a   railroad   crossing   at  the  same  time  on 
straight  tracks,  including  an  angle  of  21°  12'  (21.2°).     If  they  travel  at 
the  rate  of  40  and  50  mi.  per  hour  respectively,  how  far  apart  will  they 
be  in  45  ruin.  ? 


161.    Show  that  =  CQt  ,A      j-   Qt  ^  _  ™ 


,  .       11  Ia4-b  ,      la  —  b        2  sin  A 

162.    In  a  right  triangle  show  that  \/—    —  h\/  —   —  =  —         =• 

*a  —  b       va  +  b      y  cos  2  B 


163.    Prove  _  =  esc  A 


164.  In  any  triangle  prove  that  c  =  a  cos  J5  +  &  cos  ^4,  and  hence  show 
that  sin  (A  +  B)  =  sin  A  cos  5  4-  cos  ^1  sin  5. 

165.  Determine  the  angles  in  a  right  triangle  in  which  a  >  6,  and 
c  —  a  =  a—  &. 

166.  Prove  cos2  (a;  —  y)  —  2  cos  (#  —  y}  cos  #  cos  y  =  sin2  x  —  cos2  y. 

167.  If  sin  x  —  cos  a;  +  4  cos2  x  =  2,  find  the  ratio  of  tan  x  to  sec  x. 


168.  If  ^  +  £  =  225°,  prove  that  . 

\l  +  cot  A)  \1  -f  cot  £y     2 

169.  The  shadow  of  a.  tower  is  found  to  be  60  ft.  larger  when  the 
sun's  altitude  is  30°  than  when  it  is  45°.    Find  the  height  of  the  tower 
without  the  use  of  tables. 

170.  A  workman  is  told  to  make  a  triangular  enclosure  having  50, 
41,  and  21  yd.  as  its  sides.    If  he  malgdlthe  first  side  one  yard  too  long, 
of  what  length  must  he  make  the  other  two  sides  in  order  to  inclose 
the  required  area,  and  keep  the  perimeter  of  the  triangle  unchanged  ? 

171.  If  sin  A  is  a  geometric  mean  between  sin  B  and  cos  B,  prove 
cos  2  A  =  2  sin  (45°  -  B)  cos  (45°  -f  B). 


184  TRIGONOMETRY 

172.  If  the  diameter  of  the  earth's  orbit  about  the  sun  is  186,000,000 
miles,  and  this  diameter  when  viewed  from  the  nearest  fixed  star  sub- 
tends an  angle  of  1.52",  find  the  distance  of  the  star  from  the  earth. 

173.  In  a  circle  whose  radius  is  111.3  find  the  area  inclosed  between 
two  parallel  chords,  on  the  same  side  of  the  center  whose  lengths  are 
129.3  and  97.4. 

.  174.    If  2  tan-1  x  =  cos-1  -—  -  cos"1  -—   find  x. 

' 


175.  If  tan2  (180°  -  x)  -  sec  (180°  +  a?)  =  5,  find  cos  x. 

176.  In  order  to  fix  the  distance  between  two  islands  C  and  D,  a 
base  line,  AB,  900  ft.  long,  is  measured  on  the  shore.     Also,  Z  BAG  was 
found  to  be  110°  50'  [110.83°],  Z  DAB,  67°  51'  [67.85°],  Z  CBA,  49°  51' 
[49.85°],  ZABD,  85°  19'  [85.32°].      What  was  the  distance  between 
the  islands? 


SPHERICAL  TRIGONOMETRY 


CHAPTER   XII 
INTRODUCTION 

146.  Need  and  Utility  of  Spherical  Trigonometry.    Illus- 
trations- —  In  case  two  places  on  the  earth's    surface,  as  A. 
and  H,  have  the  same  longitude,  RS, 

and  their  latitudes,  RA  and  RH,  are 
known,  the  number  of  miles  in  the  arc 
AH  may  be  readily  determined  by 
geometry  (regarding  the  earth  as  a 
sphere).  Let  the  pupil  explain  how. 
Also  if  two  places,  as  A  and  B,  have 
the  same  latitude,  the  number  of  miles 
in  the  arc  of  a  small  circle  connecting 
them  may  be  computed  by  plane  trigonometry  (see  Art.  99). 
But  if  the  longitudes  of  two  places,  as  A  and  (7,  are  dif- 
ferent, and  also  their  latitudes,  the  number  of  miles  in  the 
arc  of  a  great  circle  AC  connecting  them  cannot  be  deter- 
mined either  by  geometry  or  plane  trigonometry.  It  can  be 
determined,  however,  by  taking  the  spherical  triangle  APC 
in  which  the  two  sides  and  the  included  angle  are  known 
(let  the  pupil  point  out  these  known  parts),  and  solving  the 
triangle  by  methods  which  are  now  to  be  considered. 

147.  Spherical  Trigonometry  is  that  branch  of  mathema- 
tics which  treats  primarily  of  the  solution  of  spherical  tri- 
angles.      It  will  be  found  that  when  any  three  of    the  six 
parts  of  a  spherical  triangle  are  given,  the  other  three  parts 

185 


186  SPHERICAL   TRIGONOMETRY 

may  be  found.  Thus  a  spherical  triangle  differs  from  a 
plane  triangle  in  that  when  three  angles  are  known  the 
three  sides  may  be  found. 

Since  a  trihedral  angle  is  closely  related  to  a  spherical 
triangle,  it  will  be  found  that  spherical  trigonometry  also 
determines  the  remaining  parts  of  any  trihedral  angle  when 
three  parts  are  given. 

Since  certain  definitions  and  principles  of  spherical  geom- 
etry are  frequently  used  in  spherical  trigonometry,  it  will  be 
useful  to  make,  at  the  outset,  a  brief  statement  of  the  lead- 
ing principles  of  spherical  geometry. 


REVIEW  OF   SPHERICAL   GEOMETRY 

148.  Definitions.  —  A  sphere  is  a  solid  bounded  by  a  sur- 
face every  point  of  which  is  equally  distant  from  a  fixed 
point  within  called  the  center.  (Every  section  of  a  sphere 
made  by  a  plane  is  a  circle.) 

A  great  circle  is  a  circle  whose  plane  passes  through  the 
center  of  the  sphere.  What  is  a  small  circle  ? 

The  axis  of  a  circle  of  a  sphere  is  that  diameter  of  the 
sphere  which  is  perpendicular  to  the  plane  of  the  circle. 

The  poles  of  a  circle  are  the  extremities  of  its  axis. 

A  spherical  triangle  is  that  portion  of  the  surface  of  a 
sphere  which  is  bounded  by  three  arcs  of  great  circles  each 
less  than  a  semicircumference. 

If  a  great  circle  be  made  to  pass  through  any  two  points  on  the  sur- 
face of  a  sphere,  the  great  circle  will  be  divided  into  two  arcs  by  the 
points.  If  these  arcs  are  unequal,  the  smaller  arc  is  less  than  a  semi- 
circumference.  It  greatly  simplifies  the  subject  of  spherical  trigonom- 
etry to  consider  only  triangles  bounded  by  arcs  each  less  than  a  semi- 
circumference  unless  there  be  some  special  reason  for  the  contrary. 

Let  the  student  define  birectangular  triangle,  trirectangu- 
lar  triangle,  quadrantal  triangle. 


INTRODUCTION  187 

A  polar  triangle  is  a  triangle  formed  by  taking  the  ver- 
tices of  a  given  spherical  triangle  as  poles  and  describing 
arcs  with  a  radius  equal  to  a  quadrant  of  a  great  circle  of 
the  sphere. 

149.  Properties  of  Points  and  Lines  on  a  Sphere. 

1.  Any  two  great  circles  of  a  sphere  intersect  at  points 
180°  apart,  i.e.  they  bisect  each  other. 

For  the  plane  of  each  of  the  two  great  circles  passes  through  the 
center,  hence  their  line  of  intersection  passes  through  the  center  and  is 
a  diameter. 

2.  The  pole  of  a  great  circle  is  at  a  quadrant's  distance 
from  each  point  on  the  great  circle. 

For  the  polar  axis  makes  a  right  angle  with  each  radius  of  the  great 
circle,  and  a  right  angle  is  measured  by  a  quadrant. 

3.  But  one  great  circle  can  be  made  to  pass  through  two 
points  less  than  180°  apart  on  the  surface  of  a  sphere. 

For  the  plane  of  a  great  circle  must  also  pass  through  the  center 
of  the  sphere,  and  three  points  determine  a  plane. 

4.  If  a  point  is'  at  a  quadrant's  distance  from  two  other 
points  on  a  sphere,  it  is  the  pole  of  the  great  circle  through 
those  points. 

150.  Properties  of   a  Spherical   Triangle.  —  In  spherical 
geometry  it  is  also  proved  that  in  any  spherical  triangle 

1.  The  sum  of  any  two  sides  is  greater  than  the  third 
side. 

2.  The  greater  side  is  opposite  the  greater  angle  and  vice 
versa. 

3.  The  sum  of  the  three  sides  lies  between  0°  and  360°. 

4.  The  sum  of  the  three  angles  lies  between  180°  and  540°. 

151.  Polar    and   Supplemental   Properties.  —  Of   especial 
importance  are  the  polar   and   supplemental   properties  of 
spherical  triangles. 


188 


SPHERICAL   TRIGONOMETRY 


opposite  in  the  other  triangle. 
Thus  in  Fig.  96,    A  +  a'  =  180°, 


1.  In  a  spherical  triangle  and  its  polar 
each  vertex  is  the  pole  of  the  side  opposite 
in  the  other  triangle. 

Thus  if  A'B'C'  be  constructed  as  the 
polar  triangle  of  ABC,  then  reciprocally 
is  ABC  the  polar  of  A'B'C'. 

2.  In  a  spherical  triangle  and.  its  polar 
each  angle  is   the  supplement  of  the  side 


£'4-6  =180°, 
<7'+c=180°. 


152.    Relation  of  the  Parts  of  a  Trihedral  Angle  to  the 
Parts  of  a  Spherical  Triangle.  —  The  planes  of  the  great  cir- 
cles which  make  three  sides  of  a  spher- 
ical triangle  meet  at  the  center  of 
the  sphere  and  thus  form  a  trihedral 
angle  whose  vertex  is  the  center  of 
the  sphere.    It  is  of  much  importance 
to  observe  the  relation  between  the 
parts  of  the  trihedral  angle  and  the 
parts  of  the  spherical  triangle. 

The  face  angles  of  the   trihedral 

angle,  viz.  AAOC,  AOB,  HOC,  are  measured  by  (i.e.  are 
equal  to,  or  contain  the  same  number  of  degrees  as)  the  sides 
or  the  spherical  triangle,  viz.  AC,  AB,  and  BC. 

Also  the  dihedral  angles  of  the  trihedral  angle  are  equal  to 
the  corresponding  angles  of  the  spherical  triangle ;  thus  the 
dihedral  angle  C-OA-B  has  the  same  measure  as  the  spherical 
angle  CAB,  viz.  the  plane  angle  2 AS  made  by  the  two 
straight  lines  TA  and  AS  tangent  to  the  arcs  AC  and  AB 
respectively,  and  therefore  perpendicular  to  the  radius  OA. 
.  Hence  the  six  parts  of  the  trihedral  angle  0  correspond  to 
the  six  parts  of  the  spherical  angle  ABC. 


FIG.  97. 


INTRODUCTION  189 

A  property  of  the  six  parts  is  sometimes  perceived  or 
derived  more  readily  from  these  parts  as  arranged  in  the 
trihedral  angle,  sometimes  more  readily  from  the  spherical 
triangle.  In  general,  it  is  more  convenient  to  obtain  methods 
of  solution  from  the  trihedral  angle;  on  the  other  hand,  the 
solution  of  problems  relating  to  the  trihedral  angle  are  usually 
obtained  more  readily  by  use  of  the  spherical  triangle. 

EXERCISE  49 

1.  If  PP1  is  the  diameter  of  a  sphere,  and  Q  any  point  on  the  sur- 
face of  the  sphere  except  P  and  P',  show  that  the  sum  of  the  arcs  PQ 
and  P'Q  is  constant. 

2.  What  must  the  sides  and  angles  of  a  spherical  triangle  be  in 
order  that  the  triangle  may  coincide  with  its  polar  ? 

3.  How  large  must  the  sides  of  a  spherical  triangle  be  in  order 
that  its  polar  lie  wholly  within  the  triangle  ? 

4.  By  use  of  cardboard  construct  a  spherical  triangle  whose  sides 
are  45°,  60°,  and  60°,  the  radius  of  the  sphere  being  3  in. 

5.  Also  (by  aid  of  cardboard  and  a  protractor)  construct  a  spheri- 
cal triangle  whose  sides  are  40°,  55°,  and  65°.     Also  construct  the  polar 
of  this  triangle. 

6.  Make  up  and  work  a  similar  example  for  yourself. 

7.  If   A,   B,   (7,   be  the  angles  of  a  spherical  triangle,  show  that 
B  +  C>  180°  -  A.     (Use  Art.  150,  4.) 

8.  Also  show  that  B  -f  C  <  180°  +  A.     (Draw  the   polar  triangle 
and  use  Art.  150,  1.) 

9.  Hence  show  that  the  spherical  excess  of  a  spherical  triangle  must 
be  less  than  twice  the  smallest  angle. 

10.  If  two  angles  of  a  triangle  are  55°  and 
110°,  find  the  maximum  value  of  the  third  angle. 

11.  If  each  of  the  legs  of  a  right  spherical 
triangle  is  less  than  90°,  prove  that  the  hy- 
potenuse and  oblique  angles  are  each  less 
than  90°. 

(SUGGESTION.  Let  ABC  be  the  right  spheri- 
cal triangle,  and  construct  the  trirectangular 
triangles  A'B'C,  AB'D,  ABE.) 


190  SPHERICAL   TRIGONOMETRY 

12.  If  the  le'gs  of  a  right  spherical  triangle  are  unlike  in  species, 
show  that  the  hypotenuse  is  greater  than  90°,  and  that  the  angle  oppo- 
site the  greater  leg  is  obtuse. 

(SUGGESTION.     Produce  the  hypotenuse  and  one  leg  to  form  a  lune.) 

13.  If  both  legs  are  greater  than  90°,  show  that  the  hypotenuse  is  less 
than  90°  and  that  both  oblique  angles  are  obtuse. 


CHAPTER   XIII 


THE   RIGHT   SPHERICAL   TRIANGLE 

153.  Trigonometric  Properties  of  the  Right  Spherical 
Triangle. —  On  a  sphere  with  center  0  and  unit  radius,  let 
ABC  be  a  spherical  tri- 
angle in  which  Z  A  CB 
is  a  right  angle.  Hence' 
plane  OBC  is  given  _L 
plane  OAC.  From  B 
draw  BD  JL  OA  and 
meeting  OA  in  D.  Also 
in  the  plane  OAC,  from 
D  draw  Z)^7  JL  CU  and 
meeting  OC  inF.  Draw 
57^.  Then  OD  is  _L  plane  DBF  (Geometry,  Art.  509). 

Hence  plane  DBF  is  _L  plane  0^4.  C  which  passes  through 
OD  (Geometry,  Art.  555).  Since  planes  DBF  and  OBC 
are  both  JL  plane  OAC,  BF  is  _L  plane  AOC  (Geometry, 
Art.  560). 

Hence  BF  is  _L  both  OF  and  DF  (Geometry,  Art.  505). 

Hence  we  have  two  right  triangles  as  follows : 


FIG.  99. 


II 


cos  C 
FIG.  100. 


D 


D  ff 

FIG.  101. 


191 


192  SPHERICAL   TRIGONOMETRY  / 

From  I,  by  Art.  41,    cos  c  =  cos  a  cos  &.  /     ...       (1) 
From  II,  by  Art.  41,  sin  a  =  sin  c  sin  A  .....       (2) 

Similarly  by  drawing  lines  from  A  instead  of  from  B 
(Fig.  99),  it  may  be  proved  that 

sin  &  =  sin  c  sin  &'?  .     .  t  .     .       (3) 
Also  from  I,  DF  =  cos  a  sin  &, 

and  from  II,  DF  =  sin  c  cos  ^1. 

By  Ax.  1,  sin  c  cos  A  =  cos  a  sin  b. 

Substituting  for  sin  b  from  (3), 

sin  c  cos  A  =  cos  a  sin  c  sin  J5. 

Whence  cos  A  =  cos  a  sin  J5.A     ...       (4) 

In  like  manner,          cos  It  =  cos  &  sin  ^l./.     ...       (5) 

From  (1),  cos  c  =  cos  a  cos  &. 

Substituting  for  cos  a  and  cos  b  from  (4)  and  (5), 

_  cos  A    cos  B 

~  7}   '  ~          7  ? 

sin  B    sin  ^L 

or  cos  c  =  cot  A  cot  J5:  .     ...       (6) 

From  (2),  sin  a  =  sin  c  sin  J_. 

Substituting  for  sin  c  and  sin  A  from  (3)  and  (5), 


-, 
sm  jfc>     cos  b 

or  sin  «  =  tan  &  cot  JB.^     ...       (7) 

Similarly,  sin  b  =  tan  a  cot  ^/  .     .     .     .       (8) 

From  (4),  cos  A  =  cos  a  sin  5. 

Substituting  for  cos  a  and  sin  ^  from  (1)  and  (3), 

,,      cos  c    sin  b 
cos  A  =  ---  •  —  —  , 

cos  b    sin  c 

or  cos  A  =  cot  c  tan  b//  .     ...       (9) 

Similarly,  cos  B  =  cot  c  tan  a/  .     .     .     ,     (10) 


THE   RIGHT   SPHERICAL   TRIANGLE 


193 


FIG.  102. 


In  the  above  proof  it  has  been  assumed  that  the  parts  of 
the  given  triangle  other  than  the  right  angle  are  each  less 
than    90°.      But    the    ten    formulas 
proved  can  be  shown  to  be  true  in  a   B 
similar  manner  when  the  parts  of  the 
triangle  are  greater  than  90°. 

For  instance,  if  the  leg  a  be  greater 
than  90°  and  b  less  than  90°,  we  have 
the  adjoining  diagram  (see  Ex.  12, 
p.  190)  in  which,  in  A  ODF, 

cos  c  =  cos  a  cos  b. 

Similarly  the  other  nine  may  be  proved  true  under  the 
given  conditions. 

As  to  the  derivation  of  the  formulas  by  use  of  the  ratio 
definitions  of  the  trigonometric  functions,  see  Art.  60. 

154.  Napier's  Rule  of  Circular  Parts.  —  The  ten  formulas 
proved  in  Art.  153  may  be  reduced  to  a  single  rule  by  the 
use  of  what  are  called  circular  parts.  The  circular  parts  of 
a  right  spherical  triangle  are  the  parts  of  the  triangle  modi- 
fied by  omitting  the  right  angle  and  taking  the  complement 
of  the  hypotenuse  and  of  the  angles  adjacent  to  it. 

Thus  use  of  the  circular  parts  in  symbols  are  co.  A,  co.  c, 
co.  B,  a,  b. 


co.JS 


FIG.  104. 


Any  one  of  the  five  circular  parts  may  be  taken  as  the 
middle  part;  the  two  parts  adjacent  to  the  part  thus  taken 


194  SPHERICAL   TRIGONOMETRY 

are  then  called  the  adjacent  parts ;  and  the  remaining  two. 
parts  are  called  the  opposite  parts. 

Thus,  if  b  is  taken  as  the  middle  part,  the  adjacent  parts 
are  co.  A.  and  a,  and  the  opposite  parts  are  co.  c  and  co.  B. 
Napier  s  Rule  for  Circular  Parts  is  then  as  follows  : 

The  sine  of  the  middle  part  =  the  product  of  the  tangents 
of  the  adjacent  parts,  or  of  the  cosines  of  the  opposite  parts. 

It  is  an  aid  in  memorizing  this  rule  to  observe  that  in  the  leading 
word  of  each  of  the  three  parts  of  the  rule,  the  first  vowel  in  the  two 
distinctive  words  is  the  same.  Thus  i  is  the  first  vowel  in  sme  and 
4fc/ddle,  a  in  tangent  and  adjacent,  and  o  in  cosine  and  opposite. 

As  an  illustration  of  the  application  of  Napier's  Rule,  if  b  be  taken 
as  the  middle  part,  we  have 

sin  b  =  tan  (co.  A)  tan  a  =  cos  (co.  c)  cos  (co.  B) 

=  cot  A  tan  a  =  sin  c  sin  B  (see  (8)  and  (3)  of  Art.  153). 

Let  the  pupil  obtain  in  like  manner  the  eight  other  formulas  of 
Art.  153,  by  taking  each  of  the  circular  parts  as  the  middle  part  in  turn. 

155.  Application  of  Napier's  Rule  to  the  Solution  of  the 
Right  Spherical  Triangle.  —  By  use  of  the  ten  formulas  of 
Art.  153  or  by  Napier's  Rule,  any  two  parts  of  a  right  tri- 
angle being  given  any  other  part  may  be  found.  In  apply- 
ing Napier's  Rule,  if  the  two  given  and  the  one  required  parts 
are  adjacent,  take  the  middle  one  of  the  three  parts  as  the  mid- 
dle part,  the  other  two  as  the  adjacent  parts. 

If  the  three  parts  are  not  adjacent,  take  the  part  standing 
alone  as  the  middle  part  and  the  other  two  of  the  three  parts 
as  the  opposite  parts. 

Ex.  1.    Solve  the  right  spherical 
•    triangle  in  which 

a  =  45°  15',  c  =  72°  30'. 

45°15/  TJ      AT       •      >     - 

By  Napier's  Rule 

cos  B  =  tan  45°  15'  cot  72°  30r. 
45°  15'  log  tan  0.00379 
72°  30'  log  cot  9.49872  -  10 


B  =  71°  27'  16"  log  cos  9.50251  - 10 


Also 


THE   RIGHT   SPHERICAL   TRIANGLE 

cos  72°  30'  =  cos  b  cos  45°  15'. 


195 


Also 


cos  45°  15' 

72°  30'  log  cos  9.47814  - 10 
45°  15'  colog  cos  0.15242 
b  =  64°  42'  51"  log  cos  9.63056  -  10 

sin  45°  15'  =  sin  A  sin  72°  30'. 
sin  45°  15' 


.*.  sin  A  = 


sin  Y. 


30'' 


45°  15'  log  sin  9.85137  - 10 
72°  30'  colog  sin  0.02058 
A  =  48°    7'  44"  log  sin  9.87195  -  10 

After  solving  any  triangle,  as  a  check  formula  use  that  formula 
involving  the  three  required  parts.  Thus,  in  the  above  example  use 
cos  71°  27'  16"  =  sin  48°  7'  44"  cos  64°  42'  50". 


Also 


48°  7'  44"  log  sin  9.87195  -  10 
64°  42'  51"  log  cos  9.63056  -  10 
71°  27'  16"  log  cos  9.50251  - 10 


Ex.  2.    In  the  right  spherical  triangle  in 

which  A  =  64.25°  and  .£  =  48.4°,  find  b. 
tyo/r' 

Taking  co.  B  as  the  middle  part  we  have 
cos  48.4°  =  sin  64.25°  cos  b, 
cos  48.4° 


hence, 


cos  b  = 


sin  64.25° 


48.4°       log  cos  9.8221  -  10 
64.25°  colog  sin  0.0454 
b  =  42.51°     log  cos  9.8675  — 10 


156.  Species  of  Parts  Found. — Where  quantities  greater 
than  90°  are  used,  in  case  any  part  given  (or  used)  is  greater, 
than  90°,  it  is  important  to  watch  the  signs  (of  the  functions) 
carefully,  since  in  the  second  quadrant  the  cos,  tan,  and  cot 
are  minus,  and  the  sine  is  plus.  If  the  cos,  tan,  Imd  cot  of  a 
computed  part  is  found  to  have  a  minus  value,  the  angle  ob- 
tained from  the  table  is  to  be  subtracted  from  180°, 


196  SPHERICAL   TRIGONOMETRY 

Ex.    Find   B   in    the   right   spherical 
triangle  in  which  a=  150°  and  c  —  80°. 

-  + 

cos  B  =  tan  150°  cot  80° 

FIG.  107.  It  is  convenient  to  write  the  sign  of   each 

factor  above  the  factor  as  is  done  above. 
Since  tan  150°  is  minus  and  co  J80°  is  plus, 

cos  B  =  product  of  a  negative  quantity  by  a  positive  quantity. 
Hence  B  is  greater  than  90°. 
Let  the  pupil  complete  the  solution. 

If  the  sine  of  an  unknown  part  is  found,  since  the  sine  of 
an  angle  and  its  supplement  are  the  same  and  both  plus,  the 
acute  angle  found  from  the  table  and  its  supplement  must 
both  be  solutions  unless  there  are  other  conditions  which 
make  one  or  the  other  of  the  solutions  impossible.  Two  of 
these  conditions  are  as  follows : 

In  any  right  spherical  triangle,  an  angle  and  the  side  oppo- 
site it  must  be  of  the  same  species }  i.e.  both  greater  than  90°, 
or  both  less  than  90°. 

For  since  sin  b  =  tan  a  cot  A. 

and  sin  b  is  always  + ,  tan  a  and  cot  A  must  be  both  +  or 
both  -. 

If  both  are  4- ,  a  and  A  are  both  less  than  90°. 

If  both  are  — ,  a  and  A  are  both  greater  than  90°. 

Hence  in  the  above  example,  since  B  is  greater  than  90°, 
AC  must  be  greater  than  90°.  Is  A  greater  or  less  than 
90°? 

Also,  in  any  right  spherical  triangle,  the  hypotenuse  is  less 
than  or  greater  than  90°  according  as  the  two  legs  are  alike 
or  unlike  in  species. 

Let  the  pupil  show  that  this  follows  from  the  formula 

cos  c  =  cos  a  cos  b. 

157.  Case  of  Two  Solutions.  —  When  the  parts  given  are  a 
side  and  the  angle  opposite  it,  there  are  two  triangles  which 


THE   RIGHT   SPHERICAL   TRIANGLE 


197 


answer    the    given    conditions. 

This  is  readily   seen    from   the 

figure.     In  the  A  ABC,  let  the    A< 

given  parts  be  the  angle  A  and  FlG  108 

BC  the  leg  opposite. 

Produce  the  unknown  sides  AB  and  AC  to  meet  at  A. 
Then  arc  ABA  =  arc  A  CA  =  180°. 

Also  Z  A'=  Z.  A  and  ^BCA=  90°. 

Hence  the  known  parts  of  the  A  A  'EC  are  the  same  as 
the  known  parts  of  ABC. 

From  the  method  of  constructing  the  figure  it  is  also 
evident  that  the  unknown  parts  of  the  one  triangle  are  the 
supplements  of  the  unknown  parts  of  the  other  triangle. 

Hence  also,  to  construct  the  two  triangles,  construct  one 
triangle  and  then  produce  the  hypotenuse  and  unknown  leg 
till  they  meet. 

Ex.  Solve  the  right  spherical  triangle  in  which  1}  =  23° 
and  ^=31°. 


Taking  a  the  middle  part,  sin  a  =  tan  23°  cot  31°. 

23°  log  tan  9.62785  -  10 
31°  log  cot  0.22123 
a  =  44°  56'  46"  log  sin  9.84808  -  10 
and  a'  =  135°  3'  14". 

Also  cos  31°  =  cos  23°  sin  A, 


sn 


and 

Also 


cos  23° 

31°  log  cos  9.93307  -  10 
23°  colog  cos  0.03597 
A  -68°  37'  15"  log  sin  9.96904  -  10 
^"  =  111°  22'  45". 
sin  23°  =  sin  c  sin  31°, 


and 


sin  31° 

23°  log  sin.  9.59188  —  10 
31°  colog  sin  0.28816 
c  =  49°  20'  44"  log  sin  9.88004  -  10 
c=  130°  39'  16". 


198  SPHERICAL   TRIGONOMETRY 

158.  Other  Special  Cases.  — Certain  special  cases  often  call 
for  special  treatment.  Thus,  if  a  and  b  are  given,  and  c  re- 
quired, and  it  is  found  that  the  value  of  c  is  close  to  either 
0°  or  180°,  this  value  can  be  found  with  greater  accuracy  by 
first  computing  A  or  B,  and  then  c.  (Why  is  this  ?) 

Also  if  a  and  c  are  given  and  b  required,  and  it  is  found 
that  the  value  of  b  is  close  to  either  0°  or  180°,  the  value  of 
b  can  be  found  more  accurately  by  use  of  the  formula 

tan2  J  b  =  tan  |(c  -  a)tan  ^(c  +  a) 

(obtained  from  the  formula  for  tan  ^b  of  Art.  70,  and  substi- 
tuting for  cos  by  from  cos  a  =  cos  b  cos  c). 

Also  in  certain  cases  the  data  of  the  problem  may  be  such 
that  a  solution  of  the  problem  is  impossible.  Thus,  if 
A  —  27°  and  B=  35°,  by  Art.  150,  4,  a  solution  is  impossible. 

A  complete  statement  of  the  conditions  which  make  the 
solution  of  a  given  triangle  impossible  is  given  in  Art.  172, 
in  those  cases  where  A  is  taken  as  equal  to  a  right  angle. 

EXERCISE   50 

Given  parts  as  follows,  solve  the  following  right  spherical  triangles, 
checking  results : 

(In  working  each  example  outline  the  work  carefully  before  looking 
up  any  logs.) 

1.  a  =  36°,  6-83°. 

2.  a  =  21°  15',  c  =  54°48'. 

3.  ^1=64°,  5  =  38°. 

4.  c  =  77°  30',  B  =  48°  18'. 

5.  a  =  20°  20'  20",  B  =  42°  6'  40". 

6.  A  =  54°  54'  42",  c  =  75°  15'  25". 

7.  A  =  115°  18'  36",  b  =  62°  18'  24". 

8.  a  =  132°  6',  b  =  77°  51'. 

9.  £  =  144°  32' 24",  c=120°. 

10.  c  =  99°  15'  36",  a  =  133°  31 '  12". 

11.  A  =  100°,  B  =  154°  37'  12". 

12.  B  =  75°  25'  12",  b  =  42°  24'. 


THE   RIGHT   SPHERICAL   TRIANGLE  199 

13.  A  =  71°,  a  =  37°J 

14.  a  -116°  44'  12",  ^=100°  16'  24"  (100.27°). 

15.  a  =  96°  18'  24 ",B  =  55°  6'  15". 

16.  a  =  56°  15',  b  =  24°  45'. 

17.  J[  =  69°  52'  36",  B  =  105°  18'  42". 

18.  b  =  16°  18',  c  =  38°  25'  48". 

19.  c  =  116°  18'  30",  A  =  50°  18'  36". 

20.  5  =63°  48',  6  =  41°  12'. 

21.  .4  =  8°  21',  5  =  87°  15' 36". 

22.  ^  =  88°  31°  12",  a  =  87°  43'  12".     Find  B  only. 

23.  A  =  89°  24'  18",  B  =  88°  31'  48".     Find  c  only. 

24.  c  =  160°  30  '36",  a  =162°  28'  48". 

25.  Why  are  we  able  to  solve  examples  like  the  preceding  in  Spheri- 
cal Trigonometry  and  not  in  Spherical  Geometry  ? 


Solve  by  use  of  four-place  tables,  having  given : 

26.  a  =  36°,  b  =  83°.  36.    A  =  100°,  B  =  154.62°. 

27.  a  =  21.25°,  c=54.8°.  37.    5=75.42°,  6=42.4°. 

28.  ^4  =  64°,  5  =  38°.  38.   A  =  71°,  a  =  37°. 

29.  c  =  77.5°,  jB  =  48.3°.  39.    a  =  116.74°,  A  =  100.27°. 

30.  a  =  20.34°,  B  =  42.11°.  40.    a  =  96.31°,  5  =  55.11°. 

31.  A  =  54.91°,  c  =  75.26°.  41.    a  =  56.25°,  6  =  24.75°. 

32.  A  =  115.31°,  b  =  62.31°.  42.   yl  =  69.88°,  B  =  105.31°. 

33.  a  =  132.1°,  6  =  77.85°.  43.    b  =  16.3°,  c  =  38.43°. 

34.  £  =  144.54°,  c  =  120°.  44.    c  =  116.31°,  .4  =  50.31°. 

35.  c  =  99.26°,  a  =  133.52°.  45.   B  =  63.8°,  6  =  41.2°. 

46.  A  =  8.35°,  B  =  87.26°. 

47.  ^1  =  88.52°,  a  =  87.72°.  Find  B  only. 

48.  ^1  =  89.405°,  B  =  88.53°.  Find  c  only. 

49.  c=  160.51°,  a  =  162.48°. 


Prove  that  in  a  right  spherical  triangle, 

50.  sin2  a  -f-  sin2  b  —  sin2  c  =  sin2  a  sin2  b. 

51.  sin2  ^1  cos2  c  =  sin2  JL  —  sin2  a. 

52.  tan  ^  (c  +a)  tan  2  (c  —  a)  =  tan2^-  b. 


200  SPHERICAL   TRIGONOMETRY 

53.  sm*-  =  sin2%os2^-f  cos2  ^  sin2! 

2i  2i  2i  a  a 

54.  tan'iB  =  ""   «-" 


sin  (c  -h  a) 

55.  sin  (c  —  6)  =  tan2  \  A  sin  (c  +  V). 

56.  sin  (c  —  a)  =  sin  b  cos  a  tan  ±  B. 

57.  sin2  A  =  cos2  .B  +  sin2  a  sin2  jB. 

58.  Prove  that,  for  any  angle  A,  tan  J-  (90°  —  -4)  =  A/-    ^4  and 

*  1  -j-  sin  A. 
hence  for  the  right  spherical  triangle  that 

tan2  (45°  -  1  A)  =  tan  1  (c  -  a)  cot  |  (c  +  a). 
When  is  this  formula  useful  in  solving  a  right  spherical  triangle  ? 

59.  Prove  that  for  the  right  spherical  triangle, 

tan2  \  B  =  sin  (c  —  a)csc  (c  -f-  a) 
and  show  when  this  formula  is  useful. 

60.  Treat  in  the  same  way  tan2  ^  c  ==  —  cos  (A  -f-  B)  sec  (A  —  B)  . 
Also  tan2  (45°  -  1  c)  =  tan  |  (A  —  a)  cot  %  (A  +  a)  . 

159.  Quadrantal  Triangles.  —  A  quadrantal  spherical  tri- 
angle is  one  which  has  one  of  its  sides  equal  to  a  quadrant. 

By  the  supplemental  property  of  spherical  triangles  (Art. 
151)  the  polar  triangle  of  a  quadrantal  triangle  is  a  right 
spherical  triangle.  Hence  to  solve  a  quadrantal  triangle, 

Solve  the  polar  triangle  of  the  given  quadrantal  triangle  and 
take  the  supplements  of  the  results. 

160.  Isosceles     Spherical     Triangles.  —  It   is    shown    in 
spherical  geometry  that  if  the  arc  of  a  great  circle  be  drawn 
from  the  vertex  of  an  isosceles  spherical  triangle  to  the  mid- 
point of  the  base,  it  will  be  perpendicular  to  the  base,  will 
bisect  the  vertex  angle,  and  divide  the  isosceles  spherical  tri- 
angle into  two  symmetrical  right  spherical  triangles.     The 
solution  of  an  isosceles  spherical  triangle  is  thus  reduced  to  the 
solution  of  a  right  spherical  triangle. 

EXERCISE  51 

Given  parts  as  follows,  solve  the  following  quadrantal  triangles, 
checking  results  : 


THE   RIGHT   SPHERICAL   TRIANGLE  201 

I/ 

1.  A  =  104°  54'  42"  [104.91°],  b  =  144°  30'  24"  [144.51°],  c  =  90°. 

2.  a  =  160°,  b  =  105°,  c  =  90°. 

3.  .4  =  115°  47'  24",  5  =  130°  31'  12",  c  =  90°. 

4.  5  =  106°  54',  6  =  100°  48'  36",  c=90°. 

In  the  following  isosceles  spherical  triangles,  given  a  =  b  and  parts 
as  follows : 

5.  a  =  56°,  c  =  99°,  find  A,  B,  C. 

6.  a  =  75°  5'  18",  A  =  35°  29'  36",  find  C,  c. 

7.  a  =  52°  30',  (7=  129°,  find  A,  c. 

8.  c  =  161°  31',  (7=  182°  24',  find  A,  a,  b. 

9.  Show  that  the  solution  of  a  spherical  polygon  may  be  reduced 
to  the  solution  of  a  right  spherical  triangle.  f 


Solve  by  use  of  four-place  tables,  having  given : 

10.  A  =  104.91°,  b  =  144.51°,  c  =  90°. 

11.  a  =  160°,  b  =  105°,  c  =  90°. 

12.  A  =  115.79°,  5  =  130.52°,  c  =  90°. 

13.  B  =  106.9°,  6  =  100.81°,  c  =  90°. 

Also  in  the  isosceles  spherical  triangle  in  which 

14.  a  =  56°,  c  =  99°,  find  A,  B,  C. 

15.  a  =  75.09°,  A  =  35.49°,  find  C,  c. 

16.  a  =  52.5°,  (7=  129°,  find  ^,  c. 

17.  c  =  161.5°,  <7  =  182.4°,  find  A,  a,  6. 


18.  In  a  quadrantal  triangle  in  which  c  =  90°,  prove  that 

tan  a  tan  6  +  sec  (7=0. 

19.  Compute  the  dihedral  angles  made  by  the  faces  of  the  five  regu- 
lar polyhedrons. 

20.  Find  the  surface  and  volume  of  a  regular  dodecahedron  whose 
edge  is  10. 

21.  If  the  side  of  a  spherical  square  is  m,  find  the  angle  M  of  the 
square. 

22.  A  marble  cutter  cuts  a  block  of  marble  with  a  rectangular  base, 
and  four  lateral  edges,  each  making  an  angle  of  45°  with  the  base  at 
its  corners.     What  is  the  dihedral  angle  between  any  two  adjacent  lat- 
eral faces  and  also  the  inclination  of  each  lateral  face  to  the  base  ? 


202  SPHERICAL   TRIGONOMETRY 

23.  Each  lateral  face  of  a  frustum  of  a  square  pyramid  makes  an 
angle  of  81°  with  the  base.     What  are  the  face  angles  at  the  corner  of 
the  base  of  the  frustum,  and  also  the  dihedral  angle  between  any  two 
adjacent  lateral  faces  ? 

24.  A  monument  has  a  rectangular  base.     One  lateral  face  makes 
an  angle  of  72°  30'  [72.5°]  with  the  base,  and  one  of  the  lateral  edges 
bounding  this  face  makes  an  angle  of  54°  48'   [54.8°]  with  the  base. 
What  angle  does  the  adjacent  lateral  face  make  with  the  first  face  ? 

25.  Find  the  dihedral  angle  made  by  any  two  adjacent  lateral  faces 
of  a  regular  twelve-sided  pyramid,  it  being  given  that  the  angle  at  the 
vertex,  made  by  two  adjacent  lateral  edges,  is  equal  to  20°. 

26.  Collect,  or  make  up,  and  work  three  examples  containing  con- 
crete applications  of   the  solution  of  right,  quadrantal,  or   isosceles 
spherical  triangles. 


CHAPTER   XIV 
OBLIQUE   SPHERICAL   TRIANGLES 

TRIGONOMETRIC  PROPERTIES   OP  OBLIQUE  SPHERICAL  TRIANGLES 

161.    Law  of  Sines.  —  In  a  spherical  triangle  the  sines  of 
the  sides  are  to  each  other  as  the  sines  of  the  angles  opposite. 


FIG.  110. 

Let  ABC  be  a  spherical  triangle  with  CD  a  perpendicular 
drawn  from  C  to  AB. 

If  this  perpendicular  fall  between  A  and  B,  as  in  Fig.  110, 
in  the  right  A  ACD,  by  Art.  154, 

Also  in  the  right  A  CDB, 

sin  p  =  sin  a  sin  B. 
.-.  by  Ax.  1,    sin  b  sin  A  =  sin  a  sin  B, 

or  sin  a  _  sin  A 

sin  b      sin  B 

In  case  the  perpendicular  CD  falls  outside  of  the  triangle 
(Fig.  Ill),  the  same  relations  are  true,  except  that  sin  Z  CBD 
is  used  instead  of  sin  B.  But 

sin  Z  CBD=  sin(180°  -B)  =  sin  B. 
Hence  the  same  result. is  obtained  as  for  Fig.  110. 

203 


204  SPHERICAL   TRIGONOMETRY 

sin  a     sin  A 


Similarly, 
and 


sin  c     sin  C  ' 

sin  b  =  sin  B 
sin  c     sin  C 


162.    Law  of  Cosines  in  a  spherical  triangle.     To  obtain 
the  relation  between  an  angle  and  the  three  sides  of  a  spheri- 
cal triangle  in  Fig.  110  of  Art.  161,  in  right  A  AC  I),  we  have 
%,  7 

COS  0  =  COS  ft  COS  X, 

and  in  right  A  BCD,  cos  a—  cosp  cos(c  —  x). 
Equating  the  values  of  cosp, 

cos  a      _cos  1} 

cos  (c  —  x)     cos  x 

-rj  cos  1)  cos(c-x) 

Hence,    cos  a  =  — 

cos  x 

_  cos  b  (cos  c  cos  x  -f  sin  c  sin  x) 

COS  X 

=  cos  b  cos  c  4-  cos  b  sin  c  tan  x.      .     .     .     (1) 
But  in  right  A  ACD, 

,  7  ,  cos  b  tan  x 

cos  ^L  =  cot  b  tan  x  = —     —  • 

sin  b 

sin  b  cos  A 


cos  6 
Substituting  for  tan  x  in  equation  (1)  above, 

7         7  .   sin  b  cos  ^4 

cos  a  =  cos  b  cos  c  +  cos  b  sin  c  -   — , 

cos  b 

or  cos  a  =  cos  &  cos  c  +  sin  &  sin  c  cos  ^4. 

Similarly, 

cos  b  =  cos  «r  cos  c  +  sin  «  sin  c  cos  1£, 
cos  c  =  cos  #  cos  b  +  sin  £«  sin  &  cos  C. 

To  obtain  the  relation  of  a  side  to  the  three  angles  of  a 
spherical  triangle. 


OBLIQUE   SPHERICAL   TRIANGLES 


205 


Let  A'B'C'  be  the  polar  A  of  ABC. 
Then  in  A  A'B'C',  by  the  property  proved 
above, 

cos  a'  =  cos  b'  cos  c'  + 

sin  &'  sin  c'  cos  A'. 
But  a!  =  180°  -A, 

&'=180°-.B,  etc. 
cos(180°  -  A)  =  cos(180°  -  B) 
cos (180°-  C)  +  sin (180°-^) sin (180°-  (7) cos (180 -a). 
Hence, 

-  cos  A  =  (  —  cos  B)(  —  cos  (7)  4-  sin  B  sin  (7(  —  cos  a), 
or  cos  A  =  —  cos  1?  cos  C+  sin  1?  sin  C  cos  a. 

Let  the  pupil  state  the  values  of  cos  B,  and  of  cos  C, 
obtained  similarly. 

163.  Formulas  for  the  Half  Angles.  —  By  the  formulas  ob- 
tained in  Art.  162,  when  the  three  sides  of  a  spherical 
triangle  are  given  it  will  be  possible  to  determine  the  angles. 
But  with  formulas  as  stated  in  Art.  162,  it  is  not  possible  to 
use  logarithms  in  the  computations.  To  obtain  formulas 
adapted  to  logarithmic  computations  we  proceed  as  follows : 
By  Art.  162, 

sin  b  sin  c  cos  A  =  cos  a  —  cos  b  cos  c. 
cos  a  —  cos  b  cos  c 


Hence,        cos  A  = 

sin  b  sin  c 

Subtracting  both  members  from  unity,  we  have 

-i  *      -i      cos  a  —  cos  b  cos  c 

1  —  cos  A  =  1  —  -   —  ;  —  -  —  — 
sin  b  sin  c 

_  cos  b  cos  c  +  sin  b  sin  c  —  cos  at 

sin  b  sin  c 
Hence,  by  Arts.  70  and  67, 


(1) 


2  sin2  1  A  = 


sin  b  sin  c 


206  SPHERICAL   TRIGONOMETRY 

Hence,  by  Art.  71, 


,_ 

sin  b  sin  c 

•  3  i   j  _si-  - 


sn     sn  c 
Denoting  the  sum  of  the  sides,  a  +  6  +  c,  by  2  s,  we  have 


and  a 

.  o  -,    ,,      sin  (s  —  &)  sin  (s  —  c) 
Hence.  sin2  1  A  =  —  •  —  '-  , 


sm  o  sm  c 


sini^  =             77~.  -     (2) 
sin  6  sm  c 

•    1  D     A/sin  (s  —  c)  sin  (s  —  a)  /Q\ 

In  like  manner,      sin  A  J3 *  V-     — —                 — ?  •     (o) 


sin  c  sin 


-  -    sm  (*      a    sm  (s  ~ 

and 


sin  a  sm 
Again,  by  adding  unity  to  both  members  of  (1),  we  have 

-,  ,  cos  a  —  cos  &  cos  c 
l-hcosJ.  =  l  +  - 

sin  6  sm  c 

_  cos  a  —  (cos  1}  cos  c  —  sin  b  sin  c) 

sin  6  sin  c 
Hence,  by  Art.  70, 

2  1    ,      cos  a  —  cos  (&  4-  c) 
2  cos2  i  J.  =  -  —  • 

sm  6  sin  c 

Hence,  by  Art.  71, 


2  ±    ,  _  2  sin  |f??  +  c  +  a)sin^-f?>  +  c  —  a) 

W  7T  -".  —  —  :      ~-.      ; 

sm  b  sm  c 
Putting  a  4-  &  +  c  =  2  s,  whence,  6  +  c  -  a  =  2  (s  -  a),  we  have 

o  -,    ,,      sin  s  sin(s  —  a) 
cosw  ^  A  =  -  —• 

sm  o  sm  c 


Or,  cos  1,1  =  V-  -•     •     •     •     (5) 

sm  b  sin  c  \ 


OBLIQUE   SPHERICAL   TRIANGLES  207 


T    vi  1   75     ^  /sin  s  sin  (s  —  6) 

In  like  manner,    cos  f  B  =  \  -         -f-    —  ,     •     • 


sin  c  sin 


-,   ~     ^  /sin  s  sin  (s  —  c) 

and  cos-J<7=\-  \        '.    •     • 

sm  a  sm  6 

Dividing  (2)  by  (5),  we  have 


tan  1  J.  =  \/sm  (s  ~  fr)  sm  (s  ~^)  -J     sin  6  sin  c 

sin  6  sin  c  sin  s  sin  (s  —  a) 

=    /sin  (s  —  b)  sin  (s  —  c)  /g\ 

sin  s  sin  (s  —  a) 


T    vi  1   r>     ^  /sin  (s  —  c)  sin  (s  —  a)  /m 

In  like  manner,    tan  ±  I>  =  \-  v  7X     ?     •      (9) 

sm  s  sin  (s  -  6) 


and,  tanl^8-     -(10) 

sin  s  sin  (s  —  c) 

164.    Formulas  for  the  Half-Sides. 

By  A1^-  1^2,     sin  ^  sin  C  cos  a  =  cos  A  +  cos  ^  cos  C. 


cos  ^L  +  cos  B  cos  6^  /n  >. 

Hence.  cosa  =  -  —  .....     (1) 

sin  B  sm  C 

rrn  -i     cos  A  +  cos  B  cos  (7 

Then,       1  -  cos  a  =  1  -  —  . 

sin  B  sm  C 

0-81          -  (cos  -5  cos  (7—  sin  ^  sin  (7)  —  cos  ^4. 
Ur,  ^  sm  ^  a  =  -  ;  —  —  —  ;  —  —  —    - 

sin  B  sm  C 

cos  (B+C)  +  cos  ^4 
sin  B  sin  (7 

Hence,    2  sm2  i  a=  - 

sin  JD  sin  6 

Denoting  the  sum  of  the  angles  ^1  +  B  4-  (7  by  2  $,  we  have 


-rj  -21  COS  >^COS  (/S'—  ^4.) 

Hence,        sm2  1  a  =  --  -A  —  —  —  L  - 

sm  B  sm  6 


Or, 


.- 
sm  ^  sin  (7 


208  SPHERICAL  TRIGONOMETRY 

In  like  manner, 


+  1     cos  S  cos  (S—B)  /Q\ 

=  \-         .          :  —  -—  ',    ....    (3) 
sin  C  sin  ^1 


.    -,        .       cos  $  cos  (S—  C)  //n 

and,  smlc=\-  -A  —  =-^  .....     (4) 

sm  A  sin  ^ 

Again,  adding  1  to  both  members  of  (1),  we  have 

cos  A  +  cos  B  cos  C 


1  +  cos  a  =  1  + 


sin  ^  sin  (7 
cos  J-  +  cos  B  cos  (7+  sin  B  sin  (7 


_ 

sin  B  sin  ( 

cos  J.  +  cos  (B—  C) 
Then,      2  cos2  i  a  =  -    —  —  „   .' 


—  „   . 
sin  B  sin  (7 

2  cosi^  +  ^-<7  cos 


sin  B  sin  (7 

,/        cos  H^  +  ^-ff]  cos  l  [- 
Or,  cos2  A  a  =  -  —  -  —  ^—  :  —  ^J- 

sin  ^  sm  (7 

But      A  +  B-C=2(S-  C),  and  .A  -  B  +  (7=  2 
Whence,    cosj  i  a  = 


- 

sm  ^  sm  C 


cos  $cos  (8—  A) 
=~cos(>g- 
In  like  manner, 


Jcos(S-B)cos(S-C) 

Or,  cosi«  =  \-          .     p   .  -V"    -^.    .     .     .    (5) 

sm  B  sm  (7 

In  like  manner, 

'S7^C.OS(f-^),    .  (6) 

sm  G  sin  ^L 


.  /COS  (A^—  A)  COS  (/S—  5)  /^x 

and,  coslc  =  \-       —  :  —  f—.  —  ^-     -^.    .     .     .    (7) 

sm  A  sm  J5 

Dividing  (4)  by  (5),  we  have, 


OBLIQUE   SPHERICAL   TRIANGLES  209 


-,  i       •%/         cos /S  cos  (/S— (7)  /lm 

and.  tan*e  =  \-  ^  y  ^  .    .     .(10) 

cos(£- J.)  cos(S-B) 

165.    Gauss's  Equations  and  Napier's  Analogies. 

Since       cos  ±  ( J.  +  B)  =  cos  (±  J.  +  \  B\ 
by  Art.  66,  cos  |(  J.  +  B)  =  cos  l  J.  cos  l  J?  -  sin  1 J.  sin  1 5. 

Substituting   for    cos  ^  J-,    cos  1  j5?    sin  1  A,    sin  ^  ^,  the 
values  obtained  in  Art.  163, 


.  Ism  s  sm  (s  —  a)  ^  Ism  s  sin  (s  — 

=\ ; — 7    v.        z\- 

sin  6  sin  c  sin  a  sm  c 


_ ^/sin  (s  —  b)  sin  (s  —  c)  ..sin  (s  —  a)  sin  (s  —  c) 
sin  b  sin  c  sin  a  sin  c 


_  sin  s  —  sin  (s  —  c)    ^/sin  (s  —  a)  sin  (s  —  b) 

sin  c  sin  a  sin  b 

But  by  Art.  71,     sin  s  —  sin  (s  —  c)  =  2  sin  -|  c  cos  (s  —  -J  c). 
Also  by  Art.  69,  sin  c  =  2  sin  \  c  cos  \  c. 


2  sin  l  c  cos  £  c 
But  *---c  =  - 


.-.  cos  l(^  +  J9)  - 

cosc 

Hence, 

cos^(^l  +  .B)  cos^c  =  cos  i(rc  +  6)  sin 
In  like  manner, 


sin      J  +  1*    cos-^  =  cos      a—  b    cos        .  ... 

cos  101  -J5)  sin  Jc=sinJ(«-l-6)  sin  |  C.   ...  Ill 
sin  |(^  -  J5)  sin  |  c  =  sin  £(«  -  6)  cos  \  C.  .     .     .    IV 
These  four  equations  are  called  Gauss's  Equations. 

Dividing  II  by  I,       tani(^  +  ^)  =  COStia7^   c<>t  l  C 


cos 
Dividing  IV  by  III,  tan±(A-B)=    g-        cot  i  <7. 


210  SPHERICAL   TRIGONOMETRY 

4 


Dividing  III  by  I,       tan  J  (a  +  6)  =       -f  tan  -|  «. 


cos 


Dividing  IV  by  II,      tan  J  (a  -  b)  =-  -*  tan  J  c. 

sin  -g-  (A.  -f-  H) 

These  equations  are  called  Napier's  Analogies. 

EXERCISE  52 

1.  In  the  first  of  Napier's  Analogies,  show-  that  tan  \(A  +  B)  and 
cos  J  (a  +  6)  must  always  have  like  signs.     Show  also  that  according 
as   a  +  b  <  180°,    =  180°,  or    >  180°,  then   A  +  B  <  180°,    =  180°,   or 
>  180°. 

2.  From  the  third  of  Napier's  Analogies,  show  that  according  as 
A  +  B  <  180°,  =  180°,  or  >  180°,  then  a  +  b  <  180°,  =  180°,  or  >  180°. 

3.  State  in  general  language  the  two  laws  of  cosines  (Art.  162). 

4.  State  in  general  language  the  results  obtained  in  Art.  163. 

5.  Also  in  Art.  164. 

6.  If  a,  6,  c  are  the  sides  of  a  spherical  triangle,  and  ar,  &',  c'  the 
sides  of  its  polar  triangle,  prove  that 

sin  a :  sin  b :  sin  c  =  sin  a' :  sin  b' :  sin  c'. 

7.  If  the  bisector  of  the  angle  A  of  the  spherical  triangle  ABC  be 
denoted  by  AD,  and  CD  be  denoted  by  b'  and  BD  by  c',  prove 

sin  b :  sin  c  =  sin  6' :  sin  c'. 

SOLUTION   OP   OBLIQUE   SPHERICAL  TRIANGLES 

166,    Cases  in  the  Solution  of  Oblique  Spherical  Triangles. 

—  Six  cases  occur  in  the  solution  of  oblique  spherical  triangles 
according  as  the  parts  given  are 

I.  Two  sides  and  the  included  angle. 

II.  Two  angles  and  the  included  side. 

III.  Three  sides. 

IV.  Three  angles. 

V.    Two  sides  and  an  angle  opposite  one  of  them. 
VI.    Two  angles  and  a  side  opposite  one  of  them. 


OBLIQUE   SPHERICAL   TRIANGLES 


211 


CASE  I.    Two  SIDES  AND  THE  INCLUDED  ANGLE  GIVEN 

167.  To  solve  Case  I  first  find  the  unknown  angles  by  the 
use  of  the  first  two  of  Napier's  Analogies  ;  the  third  side  may 
then  be  found  either  by  use  of  the  third  or  fourth  of  Napier's 
Analogies,  or  by  one  of  Gauss's  Equations. 

Which  of  the  methods  of  finding  the  third  side  involves 
the  looking  up  of  the  fewest  new  logarithms  ? 

Ex.  1.  Given  a  =  68°  20' 
25",  6  =  52°  18'  15",  (7=117° 
12'  20",  solve  the  spherical 
triangle. 

o=   68°  20' 25" 

6=   52°  18' 15" 

a  +  b  =  120°  38'  40" 

$  (a  +  b)  =60°  19'  20", 


cot  58°  36' 10". 
cot  58°  36' 10". 


£C=  58°  36'  10", 

By  the  first  two  of  Napier's  Analogies, 

cos  8°  1' 5" 


tan 


-J3)  = 


cos  60° 19' 20" 
sin  8°  1'  5" 


sin  60°  19'  20" 

8°    1'    5"  log  cos  9.99574  - 10 
58°  36f  10"  log  cot  9.78557  - 10 
60°  19'  20"  colog  cos  0.30529 
+  B)  =  50°  40'  30"  log  tan  0.08660 

8°    1'    5"  log  sin  9.14453  -  10 
58°  36'  10"  log  cot  9.78557  -  10 
60°  19'  20"  colog  sin  0.06107 
-  B)  =  5°  35'  47"  log  tan  8.99117 

Therefore,  A  =  56°  16'  47",  B  =  45°  4'  43". 

If  we  proceed  to  find  c  by  the  law  of  sines  (Art.  161),  we  shall  ob- 
tain two  values  for  c  both  greater  than  a  and  we  shall  not  know  which 
of  the  two  values  is  to  be  taken. 

Proceeding  therefore,  by  the  use  of  Gauss's  first  equation  (Art.  165). 
cog  i  c  =  cos  60°  19'  20"  sin  58°  36'  10" 
cos  50°  40' 30" 


212 


SPHERICAL   TRIGONOMETRY 


60°  19'  20"  log  cos  9.69471  - 10 
58°  36'  10"  log  sin  9.93124  - 10 
50°  40'  30"  colog  cos  0.19811 

c  =  48°  10'  17"  log  cos  9.82406  - 10 

c  =  96°  20'  34". 


CHECK.     A  log  sin  9.92000 

a  log  sin  9.96820 

9.95180 


B  log  sin  9.85009 
b  log  sin  9.89833 


C  log  sin  9.94909 

c  log  sin  9.99733 


9.95176 
43.3°,    6  = 
tri- 


9.95176 

Ex.  2.     Given    c 
19.4°,  C=  74.37°,  solve  the 
angle. 

We  obtain 

i  (a  +  6)  =31.35°, 
J  (a -6)  =11.95°, 

1(7  =  37.18°. 

By  the  first  two  of  Napier's  Analogies 
(Art.  165), 

=  cosll-?5°cot  37.18°, 
cos  31.35 

=  sinll.95°cot 
sin  31.35° 

11.95°  log  cos  9.9905 
37.18°  log  cot  0.1200 
31.35°  colog  cos  0.0686 
=  56.49°  log  tan  0.1791 
11.95°  log  sin  9.3161  - 10 
37.18°  log  cot  0.1200 
31.35°  colog  sin  0.2839 
1  (A  -  B)  =  27.69°  log  tan  0.2190  -  10 
Hence,  A  =  84.17°,  5  =  28.81°. 

By  use  of  the  first  of  Gauss's  Equations  (Art.  165), 


_ 


cos  31.35°  sin  37.18° 

cos  56.49° 
31.35°  log  cos  9.9315  -10 
37.18°  log  sin  0.7813  -10 
56.49°  colog  cos  0.2580 
c  =  20.79°  log  cos  9.9708  -  10 
c  =  41.58°. 


OBLIQUE   SPHERICAL   TRIANGLES  213 

CHECK.     A  log  sin  9.9978  B  log  sin  9.6829  C  log  sin  9.9836 

a  log  sin  9.8362  b  log  sin  9.5213  c  log  sin  9.8219 

0.1616  0.1616  0.1617 

EXERCISE  53 

Given   parts    as   follows,  solve    the    following   triangles,    checking 
results:  JL/°  /    A*l- 

1.  b  =  64°,  c  =  46'  18".  A  =  56°  24'.  ^ 

2.  a  =  73°,  c  =  47°,  B  =  113°  42'. 

3.  a  =  120°  25'  12",  b  =  80°  22'  48",  C  =  54°  33'  4".</ 

4.  6  =  61°  24'  36",  c  =  114°  37'  48",  A  =  48°  29'  24".  * 

5.  ^  =  52°  18'  24",  6  - 100°,  c  =  42°  54'. 

6.  6  =  152°  42'  36",  c  =  125°  12',  A  =  140°  37'  12". 

7.  a  =  125°  42'  40",  c  =  82°  49'  48",  5  =  59°. 


Solve  by  use  of  four-place  tables,  having  given : 

8.  b  =  64°,  c  =  46.3°,  J.  =  56.4°. 

9.  a  =  73°,  c  =  47°,  5  - 113.7°. 

10.  a  =  120.42°,  6  =  80.38°,  C  =  54.55°. 

11.  6  =  61.41°,  c  =  114.63°,  A  =  48.49°. 

12.  .4  =  52.31°,  6  =  100°,  c  =  42.9°. 

13.  b  =  152.71°,  c  =125.2°,  A  =  140.62°. 

14.  a  =  125.71°,  c  =  82.83°,  B  =  59°. 

CASE  II.     Two  ANGLES  AND  THE  INCLUDED   SIDE  GIVEN 

168.  To  solve  Case  II  first  find  the  two  unknown  sides  by 
the  use  of  the  last  two  of  Napier  s  Analogies  ;  the  third 
angle  may  then  be  found  by  the  use  of  either  the  first  or 
second  of  Napier  s  Analogies  or  by  one  of  Gauss's  Equations. 
Which  of  the  methods  of  finding  the  third  angle  involves 
the  looking  up  of  the  fewest  new  logarithms  ? 

Ex.  1.     Given  A  =  31°  40',  C=  122°  20',  b  =  40°  40',  solve 
the  triangle. 

We  have   i(C  +  A)  =  77°, 

- .4)  =45°  20', 
i  6  =  20°  20'. 


214 


SPHERICAL   TRIGONOMETRY 


By  the  last  two  %of  Napier's  Analogies, 

tan-J(c+«)  =  ^20;tan2o»20' 

COo     i    i 


FIG.  115. 


45°  20'  log  eos  9.84694  - 10 
20°  20'  log  tan  9.56887  -  10 

77°  colog  cos  0.64791 
|(c  +  a)  =  49°  11'  18"  log  tan  0.06372 

45°  20'  log  sin  9.85200  -  10 
20°  20'  log  tan  9.56887  -  10 

77°  colog  sin  0.011 28 
i (c  -  a)  =  15°  8'  8"  log  tan  9.43215  -  10 
Hence,         c  =  64°  19'  26", 
a  =  34°3'  10". 


By  Gauss's  second  equation  we  have, 

i  „      sin  77°  cos  20°  20' 
C°S^=  'cos  15°  8' 8" 

77°  log  sin  9.98872  -  10 
20°  20'  log  cos  9,97206  -  10 
15°  8'  8"  colog  cos  0.01533 
i  B  =  18°  49' 48"  log  cos  9.97611  -  10 
B  =37°  39'  36" 


CHECK.  A  log  sin  9.72014 

a  log  sin  9.74815 

9.97199 


E  log  sin  9.78600 

b  log  sin  9.81402 

9.97198 


Clog  sin  9.92683 

c  log  sin  9.95485 

9.97198 


Ex.  2.  Given  A  =  31. 67°, 
(7=  12233°,  and  6  =  40.66°, 
solve  the  triangle. 

We  have 


±(C-A)  =  45.33° 
i  b  =  20.33° 

By  the  use  of  the  last  two  of 
Napier's  Analogies, 


tan 


+  a)  = 


Cos45'3 


cos  77 


tan  20.33°, 


OBLIQUE   SPHERICAL   TRIANGLES  215 

tan  I  (c- a)  =  Sin45'33°tan  20.33°. 

sin  77 

45.33°  log  cos  9.8469-10  45.33°  log  sin  9.8520-10 

20.33°  log  tan  9.5688-10  20.33°  log  tan  9.5688-10 

77°  colog  cos  0.6479  77°  colog  sin  0.0113 

i.  (c  +  a)  =  49.18°  log  tan  0.0636          i(c  -  a)  =  15.13°  log  tan  9.4321-10 

c  =  64.31°.  a  =  34.05°. 

By  Gauss's  first  equation,  we  have 

sin  1^00*77°  cos  20.33° 

cos  49.18° 

77°  log  cos  9.3521 -10 
20.33°  log  cos  9.9721  -  10 
49.18°  colog  coa  0.1846 
$B  =  18.83°  log  sin 9.5088  -  10 
5=37.66° 

CHECK.     A  log  sin  9.7201     B  log  sin  9.7859     C  log  sin  9.9268 

a  log  sin  9.7482      b  log  sin  9.8140      c  log  sin  9.9549 

9.9719  9.9719  9.9719 


EXERCISE  54 

Given    parts   as   follows,    solve   the   following   triangles,   checking 
results :  ^ 

1.  5  =  79°,  (7=51°,  a  =  44°. 

2.  A  =  41°,  B  =  27°,  c  =  148°  30f .  > 

3.  A  =  124°  42'  36",  C  =  76°  36'  36",  b  =  48°  49'  12". 

4.  ^4  =  111°  39' .35",  5  =  127°  41'  45",  c  =  62°  47'  40". 

5.  a  =  124°  42',  5  =  106°  54',  C  =  145°  18. 

6.  C  =  133°  51',  A  =  48°  48'  36",  b  =  68°  43'. 

7.  A  =  48°  16'  48",  B  =  32°  12'  24",  c  =  116°  18'  36". 


Solve  by  use  of  four-place  tables,  having  given 

8.  jB  =  79°,  (7=51°,  a  =  44°. 

9.  .1  =  41°,  B  =  27°,  c  =  148.5°. 

10.  A  =  124.71°,  (7=  76.61°,  b  -  48.82°. 

11.  A  =  111.66°,  5  =  127.68°,  c  =  62.78°. 


216  SPHERICAL   TRIGONOMETRY 

12.  a  =  124. 7°,  B  =  106. 9°,  C=  145.3°. 

13.  C  =  133.85°,  A  =  48.81°,  b  =  68.72. 

14.  J.  =  48.28°,  B  =  32.21°,  c  =  116.31°. 

CASE  III.    THREE  SIDES  GIVEN 

169.  The  Solution  of  Case  III  may  be  effected  by  use  of 
the  formulas  of  Art.  162,  but  it  is  more  convenient  to  use 
the  formulas  for  the  half  angles  obtained  in  Art.  163.  Why 
is  this  ? 

When  it  is  required  to  compute  only  one  of  the  angles  of 
the  given  triangle,  it  is  most  convenient  to  use  the  formula 
for  the  cosine  of  the  half  angle.  Let  the  pupil  determine 
why  this  formula  is  more  convenient  than  that  for  the  sine 
or  tangent  of  the  half  angle. 

Ex.1.  Given  a=  76°  35' 36",  6  =  50°  10'  30",  c=40°0' 
10",  find  A. 

We  have  s  =  83°  23'  8",  and  s  -  a  =  6°  47'  32". 
Hence  by  Art.  163, 


LA=    /sin  83°  23'  8"  sin  6°  47'  32" 
V  sin  50°  10'  30"  sin  40°  0'  10"' 
83°  23'   8"  log  sin  9.99710  -  10 
6°  47'  32"  log  sin  9.07288  - 10 
50°  10'  30"  colog  sin  0.11464 
40°    0'  10 "colog  sin  0.19190 

2)19.37652  -  20 

i  A  =  60°  48'  8"  log  cos  9.68826  -  10 
.-.^  =  121°  36'  16". 

In  case  it  is  required  to  compute  all  three  of  the  angles  of 
the  triangle,  it  is  more  convenient  to  use  the  tangent  formula 
for  the  half  angle.  Let  the  pupil  determine  why  by  show- 
ing how  many  different  logarithms  would  need  to  be  used  in 
order  to  compute  the  three  angles  by  use  of  the  cosine  form- 
ula, and  how  many  -by  use  of  the  tangent  formula. 


OBLIQUE   SPHERICAL   TRIANGLES  217 

The  work  of  computing  all  three  of  the  angles  may  be 
further  facilitated  by  use  of  the  following  transformation : 


sm  s  sm  (8  —  a) 


-  \/sm  (6>  ~  a)  s*n  (s 
' 


sin  s  sin'2  (s  —  a) 


\/s 


sm  (s  ~  a)  sm  (s  ~ 


sin  (s  —  a)  sin  s 

Tr  .  /sin  (s  —  a)  sin  (s  —  b]  sin  (s  —  c)  , 

If  we  denote  \-  '-  by  r, 


sins 
tan  i  A  = 


sin  («  -  a) 

Likewise  tan  i  B  =  -  — , 

sin  (s  —  &) 

tan  l  C  =  - 


—  c) 

Ex.  2.     Given  a  -  124.21°,  &  =  54.3°,  c  =  97.21°,  solve  the 
triangle. 

We  have  s  =  137.86°,        s  -  b  =  83.56°, 

s  -  a  =  13.65°,          s  -  c  =  40.65°. 
Using  the  above  formula  for  r,  we  have 


y  =A/— a  13.65°  sin  83.56°  sin  40.65° 

sin  137.86° 
13.65°  log  sin  9.3729  - 10 
83.56°  log  sin  9.9973 -10 
40.65°  log  sin  9.8139  -  10 
137.86°  colog  sin  0.1733 

2)19.3574^20 
r  log  9.6787  - 10 

tani^  = r_  .'.  r  log  9. 7687 -10 

sin  13.65°  13.65°  colog  sin  0.6271 

\A  =  63.68°  log  tan  0.3058 


218  SPHERICAL   TRIGONOMETRY 

Hence  A  =  127.36°.     In  like  manner,  B  =  51.3°,  C  =  72.45°. 
Let  the  pupil  check  the  work  by  the  use  of  the  Law  of  Sines.     (See 
Ex.  1,  p.  212.) 

EXERCISE  55 

Given   parts   as   follows,    solve   the   following   triangles,    checking 
results : 

1.  a  =  52°,  b  =  37°,  c  =  43°. 

2.  a  =  150°,  b  =,125°,  c  =  43°.  \ 

3.  a  =  40°  0'  10",  b  =  50°  10'  30",  c  =  76°  Stf  36". 

4.  a  =  65°  39'  46",  6  =124°  7'  28",  c  =  159°  50'  4".  * 

5.  Given  a  =  70°  14'  20",  b  =  38°  46'  10",  c  =  49°  24'  10",  find  A. 

6.  a  =  72°  7'  12",  &  =  111°  30'  24",  c  =  44°  21'  36",  find  B. 

7.  a  =  59°  48',  &  =  115°  43'  12",  c  =  135°,  find  <7. 


By  use  of  four-place  tables  solve  the  following,  having  given : 

•8.  a  =  52°,  6  =  37°,  c  =  43°. 

9.  a  =  150°,  6  =  125°,  c  =  43°. 

10.  a  =  40.003°,  6  =  50.175°,  c  =  76.599°. 

11.  a  =  65.66°,  6  =  124.12°,  c  =  159.83°. 

12.  a  =  70.24°,  6  =  38.75°,  c  =  49.4°,  find  A. 

13.  a  =  72.12°,  b  =  111.51°,  c  =  44.36°,  find  B. 

14.  a  =  59.8°,  6  =  115.72°,  c  =  135°,  find  C. 

CASE  IV.    THREE  ANGLES  GIVEN 

170.  The  solution  of  Case  IV  is  best  effected  by  the  use 
of  the  formulas  of  Art.  164.  How  else  might  the  solution 
be  effected,  and  why  is  this  second  method  of  solution  in- 
ferior to  the  first?  In  case  but  one  side  is  required,  the 
computation  is  best  performed  by  the  use  of  th.e  formula  for 
the  sine  of  the  half  side.  Why  ? 

Since  the  cosine  in  the  second  quadrant  is  minus,  it  is 
important  in  using  the  formulas  of  this  case  to  observe  the 
sign  of  each  function  used  and  to  record  it  above  the  func- 
tion in  the  formula. 


OBLIQUE   SPHERICAL   TRIANGLES  219 

Ex.  1.    Given  A  =  58°,  B=  45°,  C=  123°,  find  c. 
We  have  S  =  113°,  S-C=-  10°. 


>•'.-' 

113°  log  cos  9.59188  - 10 
10°  log  cos  9.99335  - 10 
45°  colog  sin  0.15051 
58°  colog  sin  0.07158 

2)19.80732  -  20 

ic=    53°  13' 48"  log  sin  9.90366  =10 
c=  106°  27  '36". 

In  case  it  is  required  to  compute  all  three  of  the  sides  of 
the  triangle,  it  is  more  convenient  to  use  the  tangent  for- 
mulas. Why  ?  The  work  of  computing  all  three  of  the  sides 
may  be  further  facilitated  by  use  of  the  following  trans- 
formation : 

/          cosSvos(ti-A) 


J~  -cos  8  cos9  (8-  A) 

cos  (S-  A)  cos  (S-E)  cos  (8-  C) 


:-WI 


cos  (S-  A)  cos  (S-  B)  cos  (S-  C) 
If  we  denote 


cos  a  ,      r> 

^^A  by  E, 


cos  (S-  A)  cos  (S-B)  cos  (S-  C) 
Likewise  tan  \  b  =  -R  cos  (S  —  J5), 


Ex.  2.    Given  J.  =  20.17°,  ^=55.88°,  (7=  114.34°,  solve 
the  triangle. 

AVe  have  S  =  95.2°,  ^  -  B  =  39.32°, 

>S  -  A  =  75.03°,  £  -  C  =  - 19.14°. 


220  SPHERICAL   TRIGONOMETRY 

By  the  above  formula 


cos  95.2C 


cos  75.03°  cos  39.32°  cos  (-  19.14°) 
It  is  noted  that  the  minus  signs  compensate. 

95.2°  log  cos  8.9573  -  10 
75.03°  colog  cos  0.5878 
39.32°  cologcosO.il  14 
19.14°  colog  cos  0.0247 

2)19.6812-20 
R  log  9.8406  -  10 

tan  I  a  =  R  cos  75.03°,  R  log  9.8406  -  10 

75.03°  log  cos  9.4122  - 10 
\  a  =  10.15°  log  tan  9.2528  - 10 

Hence  a  =  20.3°.     In  like  manner  b  =  56.38°,  c  =  66.42°. 

Let  the  pupil  check  the  work  by  the  use  of  the  Law  of  Sines. 

EXERCISE  56 
Given :  / 

1.  A  =  142°,  B  =  105°,  C=  85°.     Solve  in  full. 

2.  A  =  97°  54',  B  =  106°  48'  36",  C  =  127°  35'  24".     Find  c  only/ 

3.  A  =  48°  18',  B  =  100°,  C  =  100°.     Solve  in  full.  " 

4.  A  =  73°  35'  24",  B  =  98°  7'  48",  C  =  39°  12'.     Find  a.  * 

5.  A  =  76°  30'  36",  B  =  83°  25'  48",  C  =  62°  49'  12".     Solve  in  full. 

6.  A  =  76°  29'  18",  B  =  98°  18'  36",  C  =  122°  T  42".     Find  b. 

7.  A  =  27°  30  r,  B  =  94.18°,  (7  =  83°  12 '.     Solve  in  full. 

8.  .4  =  105°  8' 10",  5  =  129°  5' 28",  (7=  142°  12' 42".     Find  c. 

9.  Show  that  it  is  impossible  to  solve  the  triangle  whose  angles  are 
142°,  125°,  and  65'. 


By  use  of  four-place  tables,  having  given 

10.  A  =  142°,  B  =  105,  C  =  85°,  solve  in  full. 

11.  ^4  =  97.9°,  B=  106.81°,  C=  127.59°,  find  c  only. 

12.  A  =  48.3°,  B  =  100°,  C=  100°,  solve  in  full. 

13.  A  =  73.59°,  B  =  98.13°,  C  =  39.2°,  find  a. 

14.  A  =76.51°,  B  =  83.43°,  C=  62.82°,  solve  in  full. 


OBLIQUE   SPHERICAL   TRIANGLES  221 

15.  A  =  76.49°,  B  =  93.31°,  C  =  122.13°,  find  b. 

16.  A  =  27.5°,  B  =  94.3°,  C  =  83.2°,  solve  in  full. 

17.  A  =  105.14°,  B  =  129.09°,  C  =  142.21°,  find  c. 

CASE  Y.   Two  SIDES  AND  AN  ANGLE  OPPOSITE  ONE  OF  THEM 

GIVEN 

171.  To  solve  Case  V  first  find  the  unknown  angle  oppo- 
site a  gwen  side  by  use  of  the  law  of  sines ;  then  find  the  third 
side  and  the  third  angle  by  use  of  Napier's  Analogies. 

172.  Number  of  Solutions  in  Case  V.  —  Under  certain  con- 
ditions  two   solutions  of   an  oblique  spherical  triangle  are 
possible. 

Thus,  if  the  given  parts  are  a,  &,  A,  and  A  is  acute  while 
a  +  b  <  180°,  b  >  90°,  and  a  <  b,  but  >  CD  (LAB),  i.e.  sin  a  > 
sin  b  sin  A9  it  may  be 
shown  that  two  solu- 
tions are  possible. 


N 


Similarly  if  A  is       X.  0/p|'  V      \ 

acute,  a  +  6>180°,    X1  '        % 


&<90°,  and  a>b, 
there  are  two  solu- 
tions. 

The  following  table  shows  the  number  of  solutions  under 
all  possible  circumstances  in  Case  V. 

I.    When  A  is  less  than  a  right  angle, 

a<b    ...........     two  solutions 

\a  =  b    .....     ......     one  solution 

0  <^  *j\j     < 

a>b  and  a  +  ~b<  180°     .....  one  solution 

a>b  and  a  +  6  =  180°  or  >  180°  .     .  no  solution 

.a<fr  ...........  two  solutions 

a  —  b  or  a  >  b  no  solution 


900 


SPHERICAL   TRIGONOMETRY 


a<b  and  a-h&<1800 
b>9Q°\a<b  and  «  +  &  =  or  >  180° 
a  =  6  or  >  b 


two  solutions 
one  solution 
no  solution 


II.    When  A  is  equal  to  a  right  angle, 

!a  <  b  or  a  =  b no  solution 

a  >  b  and  a  +  b  <  180° one  solution 

a>bsmda  +  b  =  or >  180°       .     .     .  no  solution 

a  <  b  or  a  >  b no  solution 

a  =  b infinite  number  of  solutions 

a  <  b  and  a  +  b  >  180° one  solution 

a  <  b  and  a  4-  b  =  180°  or  <  180°  .     .  no  solution 

a  =  b  or  a  >  b no  solution 


,  _ 


6  >  90 


III. 


6  =  yu 


When  ^4.  is  greater  than  a  right  angle, 

a  <  by  or  a  =  b     .....     .     .     no  solution 


a  >  b  and  a  +  b  =  180°,  or  <  180° 
a  >  b  and  a  +  b  >  180°      ... 

a  <  by  or  a  =  b 

7 

a>  b     .     . 

a  <  b  and  a  +  ~b  >  180°      ... 

a  <  b  and  a  +  b=  180°,  or  <  180° 


one  solution 
two  solutions 

no  solution 

-i  ,  . 

two  solutions 

one  solution 
no  solution 


a  =  b one  solution 

a>b two  solutions 

In  the  cases  in  which  two  solutions  are  indicated,  there 
will  be  no  solution  if  sin  a  be  less  than  sin  b  sin  A. 

It  will  be  seen  from  the  above  investigations  that  if  a  lies 
between  b  and  180°  —  b,  there  will  be  one  solution;  if  a  does 
not  lie  between  b  and  180°  —  &,  either  there  are  two  solutions 
or  there  is  no  solution  (this  does  not  include  the  cases  in 
which  a  =  6,  or  =180° -b). 

The  above  table  may  be  verified  geometrically  by  use  of 
the  accompanying  diagram.  On  the  diagram,  ED  is  the 


OBLIQUE   SPHERICAL   TRIANGLES 


223 


projection  of  a  great  circle  drawn  perpendicular  to  the  great 
circle  ADA'E. 

If  Z  A  is  acute,  it  is  represented  by  /-  PAD  (or  by  the 
equal  Z  DH'P). 

Thus,  for  example,  when  A  is  acute,  b  <  90°,  a  <  b,  we 
have  (in  general)  the  two  A  APB  and  APB'  as  solutions 
(B  and  B'  taken  in  lower  part  of  the 
diagram,  arcs  PB  and  PB'  not  being 
drawn). 

If  A  is  acute,  b  <  90°,  a  >  b,  and 
a  +  b  <  180°,  the  point  B  would  fall 
between  A'  and  jff,  and  there  would 
be  one  solution. 

If  A  is  acute,  b  <  90°,  a>b,  and 
a  +  b>  180°,  ^  would  fall  at  B  or 
j5'  in  the  upper  part  of  the  figure, 
and  there  would  be  no  solution. 

The  results  given  in  the  table  may  also  be  obtained  from 
an  analysis  of  the  formulas  used  in  the  solution  of  spherical 
triangles,  but  this  investigation  lies  beyond  the  scope  of  this 
book. 

Ex.  1.  Given  a  =  55°,  6=138°,  J.  =  42°,  solve  the  tri- 
angle. 

Since,  b  >  90°,  o <  6,  a  -+-  b  >  180°,  there  is  but  one  solution. 

By  the  law  of  sines  (Art.  161), 


sin  B  = 


sin  42°  sin  138° 


sin  55° 
42°  log  sin  9.82551 -10 
138°  log  sin  9.82551 -10 
55°  colog  sin  0.08664 

log  sin  9.73766  -  10 


The  angle  whose  log  sin  is  9.73766 
- 10  is  either  33°  8f,  or  146°  52'.  But,  since  in  the  given  triangle  the 
greater  angle  must  be  opposite  the  greater  side, 

JB=146°52'. 


224  SPHERICAL   TRIGONOMETRY 

Using  the  second  and  fourth  of  Napier's  Analogies,  we  obtain 

(7  =  54°  18' 46",  c  =  95°  59' 12". 
Let  the  pupil  check  the  work  by  the  use  of  the  Law  of  Sines. 

EXERCISE  57 

Given   parts   as   follows,    solve  the   following   spherical   triangles, 
checking  results : 

1.  ^  =  102°,  a  =  55°  24',  6  =  32°  36'X 

2.  A  =  114°  20'  14",  6  =  56°  19'  42",  a  =  66°  20'  39"/ 

3.  (7  =  44°  22'  10",  c  =  50°  45'  20",  6  =  69°  12'  40".^ 

4.  A  =  52°  18'  24*  a  =  68°  26'  36",  b  =  78°  30'  30".  V 

5.  B  =  95°  48'  36",  b  =  100°  42',  a  =  65°  27'.     Find  A 

6.  B  =  29°  18'  35",  b  =  42°,  c  =  117°  37'  12".     Find  C. 

7.  C=  129°  54',  c  =  136°  25'  12",  b  =  59°  48'.  V 


Solve  by  use  of  four-place  tables,  having  given : 

8.  A  =  102°,  a  =  55.4°,  b  =  32.6°. 

9.  A  =  114.34°,  6  =  56.33°,  a  =  66.34°. 

10.  C  =  44.37°,  c  =  50.76°,  6  =  69.21°. 

11.  A  =  52.31°,  a  =  69.44°,  6  =  78.51°. 

12.  #  =  95.81°,  6  =  100.7°,  a  =  65.45°  (find  A  only). 

13.  5  =  29.31°,  b  =  42°,  c  =  117.62°  (find  0  only). 

14.  C=  129.9°,  c  =  136.42°,  b  =  59.8°. 

CASE  VI.     Two  ANGLES  AND  A  SIDE  OPPOSITE  ONE  OF 

THEM  GIVEN 

173.  To  Solve  Case  VI  first  find  the  unknown  side  opposite 
a  given  angle  ;  then  find  the  third  side  and  third  angle  by  use 
of  Napier's  Analogies. 

In  Case  VI,  the  number  of  solutions  is  determined  by 
taking  the  polar  triangle  of  the  given  triangle  and  using 
Art.  172. 

Ex.  Given  J.  =  115°,  ^-80°,  Z>  =  84°.  Solve  the  tri- 
angle. 


OBLIQUE   SPHERICAL   TRIANGLES  225 

Since,  in  the  polar  triangle,  a'  =  65°,  b'  =  10°,  B'  =  96°,  taking  B'  as 

the  primary  angle  instead  of  A,   we  have  B'  >  90°,  «'<90°,  b'  >  a', 

a'  +  b'<  ISO0.     Hence  there  is  but  one  solution. 

By  the  law  of  sines 

sin  84°  sin  115° 

Smtt=         sin  80°  * 

84°  log  sin  9.99761 -10 
115°  log  sin  9.95728 -10 
80°  colog  sin  0.00665 


log  sin  9.96154  -  10 

There  are  two  angles  whose  log  sin  is  9.96154  — 
10,  viz. :   66°  14'  30"  and  113°  45'  30". 

Since  in  the  given  triangle  the  greater  side  must  be  opposite  the 

greater  angle, 

a  =  113°  45'  30". 

By  use  of  Napier's  Analogies,  we  find 

0  =  78°  59' 46",  c  =  82°  26' 10". 

Let  the  pupil  check  the  work. 

EXERCISE  58 

Given   parts    as    follows,    solve   the   following   spherical   triangles,' 
checking  results : 

1.  ^1  =  73°,  0=60°,  a  =  40°.  " 

2.  ^=66°24',  5  =  51°  48',  a  =  40°.l/ 

3.  B  =  148°  48',  C  =  122°  24',  c  =  75°  34'  30".  * 

4.  A  =  130°  24'  36",  C=  100°,  a  =  150°  36'  36". 

5.  C  =  36°  36'  58",  A  =  48°  23'  24",  c  =  40°  24'  36". 

6.  A  =  92°  30',  a  =  25°  42',  B  =  56°  30'. 

7.  B  =  71,  C=  120,  c  =  78°.     Is  a  solution  possible  ? 

8.  A  =  133°,  B  =  140°,  b  =  126°  (find  a  only). 


Solve  by  use  of  four-place  tables,  having  given: 
9.   .4  =  73°,  0=60°,  a  =  40°. 

10.  A  =  66.4°,  B  =  51.8°,  a  =  40°. 

11.  B  =  148.8°,  (7=122.4°,  c  =  75.575°. 

12.  A  =  130.41°,  C=  100°,  a  =  150.61°. 


226  SPHERICAL   TRIGONOMETRY 

13.  C=  36.62°,  A  =  48.39°,  c  =  40.41°. 

14.  A  =  92.5°,  a  =  25.7°,  B  =  56.5°. 

15.  B  =  71°,  C=  120°,  c  =  78°.     Is  a  solution  possible  ? 

16.  .4  =  133°,  B  =  140°,  6  =  126°  (find  a  only). 

AREA  OP   A   SPHERICAL   TRIANGLE 

174.  When  the  three  angles  of  a  spherical  triangle  are 
known,  the  area  of  the  triangle  may  be  found  by  the  follow- 
ing formula  proved  in  spherical  geometry: 


where  E  =  A  +  B  +  C-  180°  (called  the  spherical  excess). 

Ex.  1.  Find  in  terms  of  R  the  area  of  the  spherical 
triangle  in  which  J.=  78°  12'  24",  ^=68°  24'  32", 
C=  52°  35'  28". 

We  obtain  E  =  19°  12'  24"  or  69,144". 

„.      7r.fl2  19°  12'  24"  _  TT  fi2  69144" 

^80°"  648000" 

TT  log  0.49715 
69144  log  4.83975 
648000  colog  4.18842-10 
.335215  log  9.52532-  10 
.-.K=.  335215  H2  Ans. 

Ex.  2.  Find  in  terms  of  R  the  area  of  the  spherical 
triangle  in  which  A  =  78.21°,  ^=68.41°,  C=  52.59°. 

We  obtain  E  =  19.21°. 

#2  19.21° 


Hence  K  = 


180C 


TT  log  0.4971 
19.21  log  1.2835 
180  colog  7.7447-10 
.33523  log  9.5253- 10 

7T  =  . 33523 


OBLIQUE   SPHERICAL   TRIANGLES  227 

175.    When  the  three  sides  are  known,  E  may  be  computed 
by  the  formula 

tan'2  J  E  =  tan  \  s  tan  ^  (s  —  a)  tan  ^  (s  —  b)  tan  \  (s  —  c), 
called  1'Huilier's  Formula.     The  area  may  then  be  found  by 
the  method  of  Art.  174. 

Ex.  1.     Find  E  in  a  spherical  triangle  in  which  a  =144°, 
6=64°,  and  c=  133°. 

We  obtain          s  =  170°  30',  s  -  b  =  106°  30f, 

s  -  a  =  26°  30',  s  -  c  =  37°  30'. 

Hence  tan2  J  E  =  tan  85°  15'  tan  13°  15'  tan  53°  15'  tan  18°  45'. 

85°  15'  log  tan  1.08043 
13°  15'  log  tan  9.37193 
53°  15'  log  tan  0.12683 
18°  45'  log  tan  9.53078 
2)0.10997 

\  E  =  48°  37'  5"  log  tan  .05499 
E  =  194°  28'  20". 

The  proof  of  the  above  formula  is  as  follows  : 
From  the  first  of  Gauss's  Equations  (Art.  165), 

cos  %(A  +  B)  _  cos  1  (q  +  6) 
sin  ^-  C  cos  ^  c 

But  sin  l  (7=  cos  (90°  -  \  C).          (Why  ?) 

Therefore      CO8*(^-  +  -B)  _cosl(a  +  6)_ 
cos(90°-i(7)~      cosic 

Then,  by  division  and  composition, 

cos  %(A  +  B)-  cos  (90°  -  i-  C)  =  cos  ^  (a  +  b)  -  cos^  c  „. 

cos  %(A  +  B)  +  cos  (90°  -1(7)     cos^  (a  +  6)  +  cosic' 

Using  Art.  71,  and  taking  ^4.  and  B  as  any  angles,  we  have 


_tan  i(^  +  jg)tan  l^l-^).  (2) 
cos  A  4-  cos  B 

Substituting  in  (2)  for  A  and  B,%(A  +  B)  and  90°  -  1  (7, 
respectively,  and  taking  J.  and  ^5  as  any  angles,  we  have 


228  SPHERICAL   TRIGONOMETRY 

cos  I  (A  +  B}-  cos  (90°  -1(7) 
cos  I  (  J.  +  B)  +  cos  (  90°  -  1  (7  ) 


But  J&  =  J.  +  J5+  (7-180°). 


=  tani(360°-2  <7-h^L  +  £+  (7-  180°) 
=  tan  1(360°  -2 
=  tan  [90°  -£(2  (7- 
=  cot  1(2  <?-.#). 

Hence,  substituting   E  for  J.  +  5+  C-  180°,  and 
(2  C-E)im  tanl(^L  +  ,g-<7+1800),  we  have, 


iF     /QN 

Again,  substituting,  in    equation  (2),  for  A   and  I?  the 
values  -J  (a  +  &)  and  -|-  c,  and  also  substituting  s  for  -|  (a  -t-  &  +  c) 
and  s  -  c  for  -|  (a  H-  b  —  c),  we  obtain, 
cos  4  (a  4-  &)  —  cos  1-  c 


=  -  tan  1  s  tan  1  (s  -  c).    .     .     (4) 

Comparing  (1),  (3),  (4),  we  obtain, 
cotl(2(7-^)tanl^7  =  tanis  tanl(s-c).     ...     (5) 

In   like    manner   from   the   second  of  Gauss's  Equations 
(Art.  165),  we  obtain, 

tanj(2  (7-^)tanl^;  =  tanl(s-a)tanl(8'-6)  .     .     (6) 
Multiplying  (5)  by  (6),  we  have, 
tan2  J  E  =  tan  \  s  tan £(*-«)  tan \  (»  -  b)  tan^  («  -  c). 

176.    In  all  other  cases  obtain  E  by  solving  the  triangle 
and  use  the  method  of  Art.  174. 

EXERCISE  59 

1.   Find  the  number  of  square  feet  in  a  spherical  degree,  on  a  sphere 
whose  radius  is  12  feet. 


OBLIQUE   SPHERICAL   TRIANGLES  229 

2.  Find  the  length  of  an  arc  of  67°  30'  [67.5°]  in  a  circle  whose  ' 
radius  is  15  feet. 

3.  A  steamer  sailed  over  an  arc  of  6°  on  a  great  circle  of  the  earth  ^ 
in  one  day.     At  what  rate  was  the  ship  sailing  per  hour  on  the  average  ? 

4.  The  angles  of  a  spherical  triangle  are  58°,  76°  3(/  [76.5°],  and/ 
82°.     Find  the  area  of  the  triangle  if  the  radius  of  the  sphere  is  20 
inches. 

5.  Two  ships  leave  San  Francisco  at  the  same  time.     One  sails  due 
west  362  knots  and  the  other  265  knots  on  a  course  which  is  W.  41°  36' 
[41.6°]  N.,  the  first  day.     If  we  assume  that  the  ships  sail  on  arcs  of 
great  circles,  what  is  the  area  of  the  spherical  triangle  whose  vertices 
are  San  Francisco  and  the  positions  of  the  ships  at  the  end  of  the  first 
day? 

6.  Find  the  number  of  square   miles  between  the  meridians  76° 
west  and  84°  west,  if  the  radius  of  the  earth  is  3960  miles. 

7.  The  radius  of  a  sphere  is  20  inches.     Find  the  area  of  a  spheri- 
cal triangle  whose  sides  are  5.27  inches,  7.42  inches,  and  8.25  inches. 

8.  An  equilateral  spherical  triangle  on  the  earth's  surface  has  an 
area  of  1,962,000  square  miles.     Find  the  angles  of  this  triangle,  and 
also  its  perimeter  in  statute  miles. 

Find  the  areas  of  the  following  spherical  triangles,  having  given : 

9.  A=  63°  36',  B  =  95°  21'  36",  C  =  51°  48',  R  =  16  feet.      " 

10.  ^1  =  76°  29' 18",  5  =  93°  18' 36",  C=  122°  7' 42",  #  =  10  in.  * 

11.  a  =  150°,  b  =  125°,  c  =  43°,  R  =  22  meters.      * 

12.  b  =  64°,  c  =  46°  18',  A  =  56°  24',  R  =  7  inches. 

13.  A  =  41°,  B  =  27°,  c  =  148°  30',  E  =  100  meters. 

14.  a  =  109°  20'  20",  b  =  119°  29'  30",  C=  90°,  R  =  57  inches. 

15.  a  =  42°  42,  A  =  57°  48',  c  =  47°  54',  R  =  125  rods. 


By  use  of  four-place  tables,  find  the  area,  having  given: 

16.  A  =  63.6°,  B  =  95.36°,  C  =  51.8',  E  =  16  feet. 

17.  A  =  76.49°,  B  =  93.31°,  C  =  122.13°,  R  =  10  in. 

18.  a  =  150°,  b  =  125°,  c  =  43°,  R  =  22  meters. 

19.  b  =  64°,  c  =  46.3,  A  =  56.4,  R  =  7  inches. 

20.  A  =  41°,  B  =  27°,  c  =  148.5°,  R  =  100  meters. 

21.  a  =  109.34°,  b  =  119.49°,  C  =  90°,  R  =  57  inches. 

22.  a  =  42.7°,  A  =  57.8°,  c  =  47.9°,  R  =  125  rods. 


CHAPTER   XV 


SOME   APPLICATIONS   OF    SPHERICAL   TRIGONOMETRY 

177.  Finding  the  Distance  between  Two  Places  on   the 
Earth's  Surface  and  the  Direction  of  Arc  of  a  Great  Circle 
joining  Them. — If  the  earth  be  regarded  as  a  sphere,  the 
distance  between  two  places  on  its  surface  whose  latitudes 
and  longitudes  are  known,  and  also  the  direction,  or  bearing, 
of  the  line  between  them,  may  be  determined  by  Spherical 
Trigonometry. 

Let  A  and  B  be  the  two  places  on  the  earth's  surface. 
Let  the  great  circle  EGrD  be  the  equator,  and  P  its  pole, 
and  PC  and  PI)  be  the  meridians 
through  the  two  places.  Then  the  arc 
AC  is  the  latitude  of  the  place  A,  and 
BD  of  the  place  B.  Hence  in  the 
A  ABP,  PA  and  PB,  the  complements 
of  the  given  latitudes,  are  known.  Also 
the  difference  of  the  longitudes  of  the 
two  places  is  the  arc  CD  which  equals 
the  Z.  APB.  Hence  in  the  A  APB,  two  sides  and  the  in- 
cluded angle  are  known,  and  the  side  AB,  the  distance  of  the 
two  places,  may  be  determined  by  solving  the  triangle  APB. 
The  bearing  or  direction  of  the  arc  AB  is  the  angle  which 
it  makes  with  one  of  the  meridians  PC  or  PD.  Hence  the 
bearing  is  also  obtained  by  computing  the  unknown  angles 
of  the  triangle  PBA. 

178.  Reducing  any  Angle  in   Space  to  an  Angle  on  the 
Horizon.  —  Let  A  and  B  be  two  objects  observed  from  the 

230 


FIG.  121. 


SOME   APPLICATIONS  OF   SPHERICAL   TRIGONOMETRY     231 


FIG.  122. 


point  0  ;  let  OZ  be  the  vertical  line  at  0, 
OHR  the  plane  of  the  horizon,  and 
HZB  a  portion  of  the  surface  of  the 
sphere  of  which  0  is  the  center  and  OZ 
the  radius.  Let  the  angles  AOB, 
HO  A,  and  ROB  be  measured.  It  is 
required  to  determine  the  size  of  the 
angle  RON. 

Let  the   pupil   point  out  what  parts 

are  known  in  the  spherical  triangle  ZAB  and  how  HR  may 
be  determined  by  solving  the  triangle  ZAB. 

179.  Finding  the  Time  of  Day  by  an  Observation  of  the 
Sun  at  a  Place  whose  Latitude  is  known  (Declination  of  the 
Sun  being  given).  —  Let  0  be  the  position  of  the  observer, 
Z  the  zenith,  HNR  the  horizon,  P  the  pole  of  the  celestial 

sphere,  S  the  position  of  the 
sun.  When  S,  the  sun, 
crosses  the  meridian,  HZ  PR, 
it  is  noon.  Hence  the  time 
of  day  is  represented  by 
Z  ZPS,  15°  representing 
1  hour.  If  the  sun's  alti- 
tude above  the  horizon,  i.e. 
arc  SN,  be  measured,  ZS  is 
known.  PS  is  the  comple- 

ment of  the  sun's  declination  (obtained  from   the  nautical 
almanac).    How  is  arc  ZP  known  ? 
Then  how  is  Z  ZPS  obtained  ? 


180.  Finding  the  Time  of  Sun- 
rise of  a  Given  Day  of  the  Year 
at  a  Place  whose  Latitude  is 
Known.  —  Let  0  be  the  position 
of  the  observer,  Z  the  zenith, 


FIG.  123. 


FIG.  124. 


232 


SPHERICAL   TRIGONOMETRY 


HSR  the  plane  of  the  horizon,  P  the  celestial  pole,  8  the 
point  at  which  the  sun  crosses  the  horizon  at  sunrise. 
Let  the  pupil  indicate  the  method  of  the  solution. 

181.  Definitions.     Rectangular  axes   in   space   are   three 
straight  lines  each  perpendicular  to  the  other  two.     See  OX, 
OY,  OZ  in  the  next  diagram.     The  origin  is  the  point  of 
intersection  of  a  set  of  rectangular  axes,  as  0. 

A  set  of  rectangular  axes  determines  three  planes,  each  of 
which  is  perpendicular  to  the  other  two  planes.  These  three 
planes  are  called  rectangular  planes. 

182.  Property  of  a  Line  drawn  from  an  Origin  in  Space. 

— The  sum  of  the  squares  of  the  cosines  of  three  angles  which  a 
straight  line  drawn  from  an  origin  forms  with  three  rectangu- 
lar axes  equals  unity.  Let 
OX,  OY,  and  OZ  be 
three  rectangular  axes; 
let  OP  be  a  line  making 
the  angles  at  a,  ft,  y  with 
these  axes  respectively. 

Let  0  be  the  center  of 
a  spherical  surface  inter- 
secting the  axes  at  the 
points  X,  Y,  and  Z,  and 
the  line  OP  at  P. 

Then  arcs .YP,  YP,  and 
FIG.  125  ZP  measure  the  angles 

a,  ft,  y,  respectively. 

Since  Z  is  the  pole  of  the  arc  XY,  the  arc  PZ  produced 
is  perpendicular  to  XY. 

In  the  right  spherical  triangle  XPQ, 

cos  a  =  cos  XQ  cos  PQ. 
In  the  right  spherical  triangle  YPQ, 

cos  /3=  cos  YQ  cos  PQ.     Also  cos  y  =  cos  PZ. 


SOME   APPLICATIONS   OF    SPHERICAL   TRIGONOMETRY      233 

Squaring  each  equation  and  adding 
cos2  a  +  cos2  ft  +  cos2  y  =  cos2  PQ(cos2  XQ  +  cos2  YQ)  +  cos2  PZ 


183.  Properties  derived  from  Art.  182.  The  angle  which 
the  line  OP  makes  with  the  plane  XOY,  the  angle  POQ,  is 
the  complement  of  y,  the  a.ngle  which  OP  makes  with  the 
axis  OZ.  Hence  the  angles  which  the  line  OP  makes  with 
the  rectangular  planes  are  the  complements  of  the  angles. 
which  the  line  makes  with  the  rectangular  axes. 

Denoting  these  latter  angles  by  A,  13,  and  (7, 

cos  a  =  sin  A  ,         cos  ft  =  sin  B,         cos  y  =  sin  0, 
hence,  from  cos2  a  4-  cos2  ft  +  cos2  y  =  1,  we  obtain 

sin2  A  +  sin2  B  +  sin2  C=I, 
or,  in  general  language, 

The  sum  of  the  squares  of  the  sines  of  the  three  angles  which 
any  straight  line  makes  with  three  rectangular  planes  equals 
unity. 

Again,  let  a  plane  cut  three  rectangular  planes.  Let  a 
perpendicular  be  drawn  from  the  origin  to  the  cutting  plane. 
We  shall  then  have  four  planes  cutting  each  other,  and  we 
shall  also  have  four  lines,  each  of  them  perpendicular  to  one 
of  the  four  planes.  Hence  the  angle  formed  by  any  pair  of 
the  four  lines  is  equal  to  the  dihedral  angle  formed  by  the 
two  planes  to  which  these  two  lines  are  perpendicular. 
Hence,  denoting  the  angles  which  the  cutting  plane  makes 
with  the  three  rectangular  planes  by  a,  ft,  y. 

cos2  a  +  cos2  ft  +  cos2  y=l, 
or,  in  general  language, 

If  any  plane  intersect  three  rectangular  planes,  the  sum  of 
the  squares  of  the  cosines  of  the  three  angles  which  it  forms 
with  them  is  equal  to  unity. 


234  SPHERICAL   TRIGONOMETRY 

Also,  since  the  angles  which  the  cutting  plane  makes  with 
one  of  the  rectangular  planes  is  the  complement  of  the  angle 
which  the  cutting  plane  makes  with  the  rectangular  axis 
perpendicular  to  the  given  rectangular  plane,  denoting  the 
angles  which  the  cutting  plane  makes  with  the  axes  by  A, 

B,C, 

sin2  A  +  sin2  B  +  sin2  (7=1; 

or  in  general  language, 

If  a  plane  intersects  three  rectangular  axes,  the  sum  of  the 
squares  of  the  sines  of  three  angles  which  it  forms  with  them 
equals  unity. 

EXERCISE  60 

1.  Find  the  shortest  distance  on  the  earth's  surface  between  Hali- 
fax (latitude  44°  40'  K,  longitude  63° 35'  W.)  [44.66°  K,  63.58°  W.] 
and  Cape  Town  (latitude  33°  56'  S.,  longitude  18°  26'  E.)  [33.93°  S., 
18.43°  E.],  and  the  bearing  of  each  city  from  the  other. 

2.  A  ship  starting  at  a  point  whose  latitude  is  32°  42'  N.  [32.7°  N.] 
longitude  79°  53'  W.  [79.88°  W.]  sails  732  nautical  miles  in  a  north- 
easterly direction  on  a  course  which  makes  an  angle  36°  24'  [36.4°] 
with  the  meridian  at  the  starting  point.     Find  the  latitude  and  the 
longitude  of  the  final  position. 

3.  Using  the  data  of  Ex.  1,  find  the   longitude  of  a  ship  sailing 
from  Halifax  to  Cape  Town,  at  the  time  she  crosses  the  equator.     Find 
also  the  distance  in  statute  miles  from  the  point  where  the  ship  crosses 
the  equator  to  Cape  Town,  given  that  the  radkis  of  the  earth  is  3960  mi. 

4.  A   ship   sails    from    New  York   (latitude  40°  45'  N.,  longitude 
73°  58'  W.)  [40.75°  N.,  73.97°  W.]  on  an  arc  of  a  great  circle,  and  keeps 
a  course  that  is  N.  49°  29'  E.  [N.  49.48°  E.].     She  reaches  a  place 
whose  longitude  is  3° 4'  W.  [3.1°  W].     Find  the  latitude  of  this  place 
and  also  the  distance  in  statute  miles  that  the  ship  sails. 

5.  Find  the  distance  in  nautical  miles,  measured  on  the  earth's  sur- 
face, between  San  Francisco  (latitude  37°  48'  N.,  longitude,  122°  26'  W.) 
[37.8°  N.,  122.43°  W.]   and  Honolulu  (latitude   21°  18'  N.,  longitude 
157°  52'  W.)  [21.3°  N,  157.87°  W.],  and  the  bearing  of  each  city  from 
the  other. 

6.  In  what  latitude  will  a  ship  sailing  by  the  shortest  route  from 
San  Francisco  to  Honolulu  cross  the  135th  meridian  ? 


SOME   APPLICATIONS   OF   SPHERICAL   TRIGONOMETRY     235 

7.  If  two  places  are  both  in  latitude  40°,  and  their  difference  in 
longitude  is  37°,  find  in  nautical  miles  the  length  of  an  arc  of  a  great 
circle  and  also  the  length  of  a  parallel  of  latitude  joining  them. 

8.  Given,  that  the  latitude  and  longitude  of  Seattle  are  47°  36'  X. 
[47°.6°  N.]  and  122°  20'  W.  [122.33°  W.]  respectively ;  also,  the  lati- 
tude and  longitude  of  Manila  to  be  14°  35'  N.  [14.58°  N.]  and  120°  58'  E 
[120.97°  E.]  respectively,  find  the  distance  in  nautical  miles  between 
the  two  places  and  the  bearing  of  Manila  from  Seattle. 

9.  If  a  ship  sails  from  Seattle,  a  distance  of  1200  nautical  miles, 
on  a  course  that  bears  directly  for  Manila,  find  the  latitude  and  the 
longitude  of  the  point  reached. 

10.  In  Chicago,  whose  latitude  is  41°  50'  N.  [41.83°  N.],  the  sun's  alti- 
tude on  a  certain  day  is  observed  to  be  27°  30'  [27.5°].     Given  that  the 
sun's  declination  is  13°  N.  and  that  the  observation  is  made  in  the  fore- 
noon, what  is  the  time  of  day  ? 

11.  At  what  hour  will  the  sun  rise  in  Quebec  (latitude  46°  48'  N.) 
[46.8°  N.]  if  its  declination  is  17°  36'  N.  [17.6°  N.]  ? 

12.  At  what  time  will  the  sun  set  in  New  Orleans  (lat.  29°  58'  N.) 
[29.97°  N.]  if  its  declination  is  14°  S.  ? 

13.  Given  A  and  B  two  points  in  space  above  the  horizon,  0  a  point 
on  the  plane  of  the  horizon,  OA'  and  OB',  the  projections  of  OA  and 
OB  respectively  on  this  plane.     Angle  A'OA  =  22°  24'  [22.4°],  angle 
B' OB  =  63°  42'  [63.7°],  and  angle  .405=82°  48'  [82.8°],  find  the  angle 
A' OB'. 

14.  If  the  declination  of  the  sun,  on  April  1,  is  4°  30'  N.  [4.5°  W.], 
find  the  time  of  sunrise  in  Melbourne  if  the  latitude  of  Melbourne  is 
37°  50'  S.  [37.83°  S.]. 

15.  Given   that   the  latitude  and  longitude  of  Chicago  are  41°  50' 
[41.83°]  and  87°  37'  [87.61°]  respectively,  and  also  that  the  earth's 
radius  is  3960  miles,  find : 

(1)  The  distance  of  Chicago  from  the  plane  of  the  equator. 

(2)  Its  distance  from  the  earth's  axis. 

(3)  Its  distance  from  the  plane   passing   through  the  meridian  of 
Greenwich. 

(4)  The  circumference  of   a  circle  that  it  describes  every   day   in 
consequence  of  the  earth's  rotation  on  its  axis. 

16.  Taking  the  latitude  of  Quebec   as  found  in  Ex.  11,  and  the 
declination  of  the  sun  as  12°  S.,  at  what  hour  in  the  afternoon  will  the 
sun  have  an  altitude  of  24°  42'  [24.7°]  ? 


236  SPHERICAL   TRIGONOMETRY 

17.  Determine  the  length  of  the  longest  day  of  the  year,  at  a  place 
whose  latitude  is  40°  N.       (In  this  example  the  sun's  declination  is 
taken  as  23°  30'  K  [23.5°  N.J,  or  the  distance  of  the  Arctic  Circle  from 
the  north  pole.) 

18.  Let  the  line  OP  meet  the  axes  OX,  OF,  OZ,  so  that  ZXOP=70°, 
ZZOP=60°;  find  TOP.     Also  find  the  angles  which  OP  makes  with 
the  three  rectangular  planes. 

19.  In  Ex.  18,  let  OP  be  18.275  in.   long.      Find    the   lengths    of 
the  perpendiculars  let  fall  from  P  on  the  three  axes ;    also  find  the 
lengths  of  the  projections  of  OP  on  the  three  axes. 

20.  Also  find  the  lengths  of  the  perpendiculars  from  P  on  the  three 
rectangular  planes ;  also  find  the  lengths  of  the  projections  of  OP  on 
the  three  planes. 

21.  A  plane  cuts  the  three  rectangular  planes,  and  a  line  OP  is  per- 
pendicular to  the  cutting  plane.     OP  makes  an  angle  of  48°  with  the 
plane  XOZ,  and  33°  with  the  plane  YOZ.     What  angle  does  OP  make 
with  the  plane  X 0  Y.     Also  what  angles  does  the  cutting  plane  make 
with  the  three  axes  and  with  the  three  rectangular  planes  ? 

22.  Why  are  we  able  to  get  results  like  these  by  Spherical  Trigo- 
nometry and  not  by  Spherical  Geometry  ? 

23.  Show  in  what  sense  the  property  demonstrated  in  Art.  182  is 
true  for  any  line  in  space  which  is  parallel  to  line  OP? 

24.  If  the  edges  of  a  parallelepiped  be  denoted  by  a,  b,  c,  and  the 
plane  angles  formed  by  these  edges  be  denoted  by  a,  ft,  y,  and  the 
volume  by  V,  show  that 

V=  abc  Vl  —  cos2  a  —  cos2  ft  —  cos2  y  +  2  cos  a  cos  ft  cos  y. 

25.  Also  obtain  a  formula  for  the  length  of   the  diagonal  of  the 
parallelepiped  of  the  preceding  example. 

26.  In  a  regular  polyhedron,  if  n  denote  the  number  of  sides  in  each 
face,  p  the  number  of  face  angles  in  each  polyhedral  angle,  and  i  the 
inclination  of  two  adjacent  faces,  prove  that 

sin  i  i  =  cos  -  esc—  . 
p        n 

27.  Also  if  the  edge  of  a  regular  polyhedron  be  denoted  by  a  and 
the  number  of  faces  by  N  and  the  volume  by  F,  show  that 

V=  2*4  Nna?  cot2  -tan  |  i. 


SOME   APPLICATIONS   OF   SPHERICAL   TRIGONOMETRY     237 

28.  Make  up  and  work  an  example  concerning  the  distance  between 
two  places  on  the  earth's  surface  when  the  bearing  between  them  is 
known,  and  also  the  latitudes  of  the  two  places  and  the  longitude  of  one 
of  them. 

•    29.    Given  the  time  of  day,  show  how  to  find  the  latitude  of  a  place 
by  an  observation  of  the  sun,  the  declination  of  the  sun  being  known. 

30.  Collect,  make  up,  and  work  as  many  examples  as  you  can,  each 
containing  a  different  practical  application  of  the  solution  of  oblique 
spherical  triangles. 


LOGARITHMIC  AND 
TRIGONOMETRIC  TABLES 


EDITED   BY 

FLETCHER   DURELL,   Pn.D. 

HEAD    OF    THE    MATHEMATICAL    DEPARTMENT 
THE    LAWRENCEVILLE    SCHOOL 


NEW   YORK 
CHARLES   E.   MERRILL   CO. 

44-60  EAST  TWENTY-THIRD  STREET 

1911 


DurelFs  Mathematical  Series 

Plane  Geometry 

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Plane  Trigonometry 

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Plane  Trigonometry  and  Tables 

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Tables 

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Plane  and  Spherical  Trigonometry,  with 
Tables 
351  pages,  8 vo,  cloth $1.40 

Plane  and  Spherical   Trigonometry,  with 
Surveying  and  Tables 

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Logarithmic  and  Trigonometric  Tables 

114  pages,  8vo,  cloth 75  cents 

Copyright,  1910,  by  Charles  E.  Merrill  Co. 
[3] 


CONTENTS 

PAGE 

INTRODUCTION  TO  TABLES 5 

TABLES  : 

I.     FIVE-PLACE  LOGARITHMS  OF  NUMBERS  1-10,000  ...      21 

II.     LOGARITHMS  AND  COLOGARITHMS  OF  MUCH-USED  NUMBERS      40 

UI.     FIVE-PLACE  LOGARITHMS  OF  THE  SINE,  COSINE,  TANGENT, 

AND  COTANGENT  FOR  EACH  MINUTE  OF  THE  QUADRANT      41 

IV.     AUXILIARY  FIVE-PLACE  TABLE  FOR  SMALL  ANGLES    .        .       87 

V.  FOUR-PLACE  TABLE  OF  THE  NATURAL  SINE,  COSINE,  TAN- 
GENT, AND  COTANGENT  FOR  EVERY  TEN  MINUTES  OF 
THE  QUADRANT  .  .  .......  91 

VI.     FOUR-PLACE  LOGARITHMS  OF  NUMBERS  1-2000     ...      97 

VII.  FOUR-PLACE  LOGARITHMS  OF  THE  TRIGONOMETRIC  FUNC- 
TIONS FOR  ANGLES  OF  THE  QUADRANT  EXPRESSED  BY  THE 
DECIMALLY  DIVIDED  DEGREE 103 

VIII.     CONVERSION    OF   MINUTES    AND    SECONDS    INTO    DECIMAL 

PARTS  OF  A  DEGREE 114 

IX.     CONVERSION    OF    DECIMAL    PARTS    OF     A    DEGREE    INTO 

MINUTES  AND  SECONDS  .        .        .     114 


INTRODUCTION   TO    TABLES 

1.  Number  of  Decimal  Places  in  Tables.     All  trigonometric 
work  is  based   on  the  results  of    measurements.       But  no 
measurement  is  accurate  beyond  the  sixth  or  seventh  figure; 
this  is  owing  to  the  limitations  of  our  eyesight  and  sense  of 
touch-perception,  and  to  the  ultimate  imperfections  in  all  our 
instruments  of  measurement. 

Thus  a  mile  (63,360  inches)  can  be  measured  to  within  y1^  inch  of  its 
true  length ;  an  inch  can  be  measured  only  to  within  a  millionth  part 
of  itself,  etc.  So  great  a  degree  of  accuracy,  however,  can  be  obtained 
only  by  applying  every  possible  refinement  of  accuracy.  Ordinary 
measuring,  such,  for  instance,  as  that  done  by  a  carpenter,  is  accurate 
only  to  the  second  or  third  figure,  that  is,  to  within  y^  or  T^^  part. 
Hence  it  would  be  absurd  for  a  carpenter  or  surveyor  to  use  a  number 
like  7.382654  ft. ;  7.38+  ft.  is  sufficient. 

In  6,543,786,  if  the  figure  6  to  the  right  is  -|  inch  long,  how  long 
would  the  figure  6  on  the  left  be  if  its  length  were  made  proportional  to 
its  value? 

Hence  four-place  tables  are  sufficiently  accurate  for  all  or- 
dinary work  (such  as  is  done  by  a  land  surveyor,  or  in  a  physi- 
cal laboratory  under  ordinary  circumstances).  Five-place 
tables  give  all  the  accuracy  required  except  in  very  rare  cases, 
when  six-  or  seven-place  tables  may  be  used.  But  the  latter 
cases  are  beyond  the  scope  of  this  book. 

TABLE  I.    FIVE-PLACE  LOGARITHMS  OF  NUMBERS  1-1O,OOO  (pp.  21-39) 

2.  General  Description  of  Table  I.     Table  I  consists  of  two 
parts.     Part  I  occupies  p.  21  and  gives  the  logarithms  (both 
characteristic  and  mantissa)  of   numbers    1-100.      Part  II 
occupies  pp.  22-39,  contains  mantissas  only,  and  gives  these 
for  all  numbers  from  1  to  10,000. 

5 


6  TRIGONOMETRY 

In  using  Part  II  the  characteristic  of  each  logarithm  must  be  deter- 
mined and  supplied  in  accordance  with  the  methods  stated  in  Arts.  4 
and  5  of  Durell's  Plane  Trigonometry. 

DIRECT  USE  OF  TABLE  I 

3.  To  find  the  mantissa  for  a  number  containing  four  figures. 

In  the  given  table  the  left-hand  column  (headed  N)  is  a 
column  of  ordinary  numbers.  The  first  three  figures  of  the 
given  number  whose  mantissa  is  sought  are  found  in  thl§ 
column.  In  the  top  row  of  each  page  are  the  figures 
0,  1,  2,  3,  4,  5,  6,  7,  8,  9.  The  fourth  figure  of  the  given 
number  is  found  here. 

Hence,  to  obtain  the  mantissa  of  3647,  for  instance,  we  take  364  in 
the  first  column  on  page  27  and  look  along  the  row  beginning  with  364 
till  we  come  to  the  column  headed  7.  The  mantissa  thus  obtained  is 
.56194. 

The  first  two  figures  of  the  row  of  mantissas,  viz.  56,  are 
supposed  to  be  repeated  in  connection  with  each  mantissa 
that  follows  till  another  complete  mantissa  is  given.  The 
use  of  a  *  indicates  that  the  first  two  figures  of  the  mantissa 
are  to  be  taken  from  the  beginning  of  the  line  of  mantissas 
which  follows. 

Thus,  the  mantissa  of  1125  is  .05115,  not  .04115. 

If  the  number  whose  mantissa  is  sought  contains  less  than 
four  figures,  in  using  the  tables  we  regard  enough  zeros  as 
annexed  to  the  given  figures  to  make  up  four  figures.  In 
Chapter  I  of  Durell's  Plane  Trigonometry  it  is  shown  that 
doing  this  does  not  affect  the  mantissa. 

Thus,  to  find  the  mantissa  of  271,  we  find  the  mantissa  of  2710,  viz. 
.43297. 

Similarly  the  mantissa  of  7  is  the  same  as  that  of  7000,  viz.  .84510. 

4.  To  find  the  mantissa  of  a  number  containing  five  or  six 
figures.     Interpolation.     The  method  consists  in  finding  the 
mantissa  for  the  first  four  figures  and  adding  a  correction  for 


INTRODUCTION   TO   TABLES  7 

the  fifth,  or  for  the  fifth  and  sixth  figures.  This  correction 
is  computed  on  the  assumption  that  the  differences  in  loga- 
rithms are  proportional  to  the  differences  in  the  numbers  to 
which  they  belong.  Though  this  proportion  is  not  strictly 
accurate,  it  is  sufficiently  accurate  for  practical  purposes. 

Ex.   Find  the  mantissa  of  1581.47. 

m.  for  1582  =  .19921  Mantissa  of  1581  =  .19893 

m.  for  1581  =  .19893  .00028  x  .47          =  .00013 

Diff.  for  1  =  .00028  Mantissa  of  1581.47  =  .19906,  Ans. 

For  since  an  increase  of  1  in  the  number  makes  an  increase  of  .00028 
in  the  mantissa,  an  increase  of  .47  in  the  number  will  make  an  increase 
of  .47  of  .00028,  that  is,  of  .00013  in  the  logarithm. 

As  in  the  mantissa,  so  in  the  correction  only  five  places  of  figures 
may  be  used.  If  the  figure  in  the  sixth  place  of  the  correction  is  5  or 
a  larger  number,  the  figure  in  the  fifth  place  of  the  correction  is  to  be 
increased  by  1;  if  less  than  5,  the  figures  after  the  fifth  place  are  to  be 
rejected.  Thus  if  the  above  correction  had  been  .000135  it  would  have 
been  treated  as  .00014.  If  it  had  been  .OOOJ346  it  would  have  been 
treated  as  0.00013. 

The  difference  between  the  mantissas  of  two  successive 
numbers  is  called  the  tabular  difference. 

Hence,  in  general,  to  find  a  mantissa  for  a  number  contain- 
ing five  or  six  figures: 

Obtain  from  the  table  the  mantissa  for  the  first  four  figures, 
and  also  that  for  the  next  higher  number,  and  subtract; 

Multiply  the  difference  between  the  two  mantissas  by  the  fifth 
figure  (or  fifth  and  sixth  figures)  expressed  as  a  decimal, 
and  add  the  result  to  the  mantissa  for  the  first  four  figures. 

5.    Hence,  to  find  the  log  of  a  given  number: 

Determine  the  characteristic  by  Art.  4  or  5,  Chapter  I; 
Neglect  the  decimal  point  (in  the  given  number)  and  obtain 
from  the  table  the  mantissa  for  the  given  figures. 


TRIGONOMETRY 


m.  of  7855  =  .89515  log  .07854  =  8.89509  -  10 

m.  of  7854  =  .89509  .00006  x  .6  =    .00004 


Ex.  1.    Find  log  3.62057. 

m.  of  3.621  =  .55883  log  3.620  =  0.55871 

m.  of  3.620  =  .55871  .00012  x  .57  =    .00007 

.00012  log  3.62057  =  0.55878,  Ans. 

Ex.  2.    Find  log  .078546. 

loa-  .07854  =  8.89509  -  10 

.00006  log  .078546  =  8.89513  -  10,  Ans.  \ 

For  examples  to  be  worked  by  the  pupil,  see  the  first  part 
of  Exercise  3  D£  Durell's  Plane  Trigonometry. 

INVERSE  USE  OF  TABLE  I     . 

6.  To  find  an  antilogarithm,  that  is,  to  find  the  number 
corresponding  to  a  given  logarithm. 

Since  the  characteristic  depends  only  on  the  position  of 
the  decimal  point  and  not  on  the  figures  forming  the  given 
number,  the  characteristic  is  neglected  at  the  outset  of  the 
process  of  finding  the  antilogarithm. 

(a)  If  the  given  mantissa  can  be  found  in  the  table : 
Take  from  the   table  the  figures  corresponding  to  the  man- 
tissa of  the  given  logarithm; 

Use  the  characteristic  of  the  given  logarithm  to  fix  the  deci- 
mal point  in  the  number  obtained  from  the  table. 

Ex.  1.    Find  the  antilogarithm  of  1.44138. 

The  figures  corresponding  to  the  mantissa  .44138  are  2763.  Since 
the  characteristic  is  1,  there  are  two  figures  at  the  left  of  the  decimal 
point. 

Hence  the  antilog  1.44138  =  27.63. 

Or,  if  log  x  =  1.44138,  x  =  27.63. 

(b)  In  case  the  given  mantissa  does  not  occur  in  the  table : 

Obtain  from  the  table  the  next  lower  mantissa  with  the  corre- 
sponding four  figures  of  the  antilogarithm; 


INTRODUCTION   TO   TABLES  9 

Subtract  the  tabular  mantissa  from  the  given  mantissa; 

Divide  this  difference  by  the  difference  between  the  tabular 
mantissa  and  the  next  higher  mantissa  in  the  table; 

Annex  the  quotient  to  the  four  figures  of  the  antilogarithm 
obtained  from  the  table; 

Use  the  characteristic  to  place  the  decimal  point  in  the  result. 


C> 


x.  1.    Find  the  antilog  of  2.42376. 


The  mantissa  .42376  does  not  occur  in  the  table,  and  the  next  lower 
mantissa  is  .42374.  The  difference  between  .42376  and  .42374  is 
.00002. 

If  a  difference  of  16  in  the  last  two  figures  of  the  mantissa  makes 
a  difference  of  1  in  the  fourth  figure  of  the  antilog,  a  difference  of  2  in 
the  last  figure  of  the  mantissa  will  make  a  difference  of  T2¥  of  1  or  .125 
(or  -13)  with  respect  to  the  fourth  figure  of  the  antilog.  Hence  we 
have 

antilog  2.42376  =  265,313-     Ans. 

374 

16)2.00(.13~ 
16 
40 

Ex.2.    If  log  x  =  7.26323-10,  find  x. 

Nearest  less  mantissa  =  .26316,  whose  number  is  1833. 
Tab.  diff.  =  24.  7-*-24  =  .29+.  Hence  a;  =.00183329,  Ans. 

The  first  part  of  Exercise  4  of  Durell's  Plane  Trigonom- 
etry should  be  worked  at  this  point. 

TABLE    II.       LOGARITHMS     AND     COLOGARITHMS     OP     MUCH-USED 

NUMBERS     (p.  4O) 

This  table  explains  itself. 

TABLE   III.     FIVE-PLACE   LOGARITHMS    OP    TRIGONOMETRIC    FUNC- 
TIONS FOR  EVERY  MINUTE  OF  THE  QUADRANT  (pp.  41-86) 

7.    Description  of  Table  III.     This  table  gives  the  loga- 
rithms of  the  sine,  cosine,  tangent,  and  cotangent  of  each 
*-*•>   minute  of  angle  from  0°  up  to  90°. 


10  TRIGONOMETRY 

Where  — 10  is  a  part  of  the  characteristic  of  the  log  function  it  is 
omitted  for  the  sake  of  economy  of  space.  This  omission  occurs  at  the 
end  of  the  log  function  of  each  angle  except  for  log  tangents  from  45° 
to  90°,  and  log  cotangents  from  0°  to  45°. 

For  angles  between  0  and  45°,  the  required  functions  are 
printed  at  the  top  of  the  columns,  the  number  of  degrees  at 
the  top  of  the  page,  and  the  number  of  minutes  in  the 
hand  column. 

For  angles  between  45°  and  90°,  the  required  functional 
printed  at  the  bottom  of  the  columns,  the  number  of  degrees 
at  the  bottom  of  the  page,  and  the  number  of  minutes  in  the 
right-hand  column. 

Thus, 

log  sin  26°  37'  =  9.65130  - 10  (p.  68).     log  tan  67°  48'  =  0.38924  (p.  64). 
log  sin  58°  16'  =  9.92968  - 10  (p.  73).     log  cot  12°  23'  =  0.65845  (p.  54). 

Let  the  pupil  determine  why  each  column  of  the  table 
has  the  name  of  a  trigonometric  function  at  the  top  and  the 
name  of  the  corresponding  co-function  at  the  bottom  of  the 
column. 

Let  him  also  determine  why  — 10  is  to  be  annexed  at  the 
end  of  some  log  trigonometric  functions  as  taken  from  the 
tables,  and  not  at  the  end  of  others. 

» 

DIRECT  USE  OF  TABLE  III 

8.  Given  the  degrees,  minutes,  and  seconds  of  an  angle,  to 
find  a  logarithmic  trigonometric  function  of  the  angle.  After 
finding  the  log  function  for  the  given  number  of  degrees  and 
minutes,  the  log  function  for  the  given  number  of  degrees, 
minutes,  and  seconds  is  found  by  interpolation. 

Ex.  1.    Find  the  log  sin  37°  42'  53". 

The  log  sin  37°  42'  is  9.78642,  and  the  difference  between  this  and 
log  sin  37°  43'  is  16- 

Since  an  increase  of  1'  in  the  angle  makes  an  increase  of  16  in  the 


INTRODUCTION   TO   TABLES  11 

last  two  places  of  the  log  sin,  an  increase  of  53"  or  |~|  of  1'  will  make 
an  increase  of  ||  of  16  in  the  log  of  the  function. 

Hence  we  have 

log  sin  37°  42'=  9.78642  -  10 

Diff.  for  53"  =  ff  of  16=  14 

log  sin  37°  42'  53"  =  9.78656  -  10 

x.  2.   Find  the  log  sin  53°  27'  18". 

log  sin  53°  27'  =  9.90490  -  10 
Diff.  for  18"  =  M  of  9  =  3 


log  sin  53°  27'  18"  =  9.90493  - 10 

Ex.3.    Find  log  cos  23°  48' 12". 

Since  the  cosine  of  an  angle  decreases  as  the  angle  increases,  the 
log  of  23°  49'  is  less  than  the  log  cos  23°  48'.  Hence  the  correction  for 
12"  must  be  subtracted  from  the  log  cos  23°  48'. 

Thus  log  cos  23°  48'  =  9.96140  - 10 
Diff.  for  12"  =  1|  of  5  =  1 

log  cos  23°  48'  12"  =  9.96139 -10 

Ex.  4.   Find  log  cot  57°  18'  43". 

log  cot  57°  18'  =  9.80753  - 10 
Diff.  for  43"  =  28  x  £ £  =  20 

log  cot  57°  18'  43"  =  9.80733  - 10 

Hence,  in  general, 

Obtain  from  the  table  the  log  function  for  the  given  number 
of  degrees  and  minutes; 

Also  obtain  from  the  table  the  log  function  for  the  angle,  1 
minute  greater;  find  the  difference  between  these  two  log  func- 
tions; multiply  this  difference  by  -  —;  this  will  give  the 
correction  for  seconds; 

Add  the  correction  for  seconds  in  case  of  sine  and  tangent 
(direct  functions') ; 

Subtract  the  correction  in  case  of  cosine  and  cotangent  (com- 
plementary functions'). 


12  TRIGONOMETRY 

9.    Log  Secants.     To  find  the  log  secant  of  an  angle,  use 

the  formula  sec  x  —  — -  —     .*.  log  sec  x  =  0  +  colog  cos  x. 
cosx 

Thus  log  sec  39°  28'  23"  -  colog  cos  39°  28'  23". 

But  log  cos  39°  28'  23"  =  9.88757  -  10. 
colog  cos  39°  28'  23"  or  log  sec  39°  28'  23"  =  0.11243. 

10.  Log  Functions  of  Angles  greater  than  90°.      ByfllB 
methods  of  Chapter  IV,  a  trigonometric  function    of   aiy 
angle  greater  than  90°  can  be   reduced  to  a  trigonometric 
function  of  an  angle  less  than  90°. 

Thus,  since  sin  A  =  sin  (180°  -  A), 

sin  113°  27'  =  sin  66°  33'. 
.-.  log  sin  113°  27'  =  log  sin  66°  33'  =  9.96256  - 10. 

Also  cos  A  =  -  cos  (180°  -  A). 

Hence,  log  cos  A  =  log  cos  (180°  -  A)(ri),  the  small  n  being 
annexed  to  show  that  the  function  whose  log  is  being  used 
is  a  negative  quantity. 

Thus  log  cos  142°  18'  =  log  cos  37°  42'  (n)  =  9.78642  -  10  (n). 

At  this  point  work  the  first  part  of  Exercise  14  of  Durell's 
Plane  Trigonometry. 

INVERSE  USE  OF  TABLE  III 

11.  Given  the  logarithm  of  a  function  to  find  the  correspond- 
ing acute  angle  (or  find  antilog  sin,  antilog  cos,  etc.  or  /.log 

'  sin,  /.log  cos,  etc.)  Obtain  from  the  table,  if  possible,  the 
number  of  degrees  and  minutes  corresponding  to  the  given 
logarithmic  function. 

Ex.  If  log  tan  A  =  9.92535  -  10,  find  the  angle  A. 

By  consulting  the  table,  tangent  column,  we  find  that  ^4  =  40°  6'. 
Or  antilog  tan  9.92535  -  10  =  40°  6'. 

If  the  given  logarithmic  function  does  not  occur  in  the 
table : 


INTRODUCTION   TO   TABLES  13 

Obtain  from  the  table  the  next  less  logarithm  of  the  same  func- 
tion, noting  the  corresponding  number  of  degrees  and  minutes; 
subtract  this  logarithm  from  the  given  logarithm; 

Divide  the  difference  so  obtained  by  the  tabular  difference  for 
V  and  multiply  by  60";  the  result  will  be  the  correction,  in 
seconds,  to  be  added  in  case  of  sine  and  tangent,  and  sub- 
fw^  in  case  of  cosine  and  cotangent,  to  the  angle  already 

n 

1 

Ex.  1.   Find  antilog  sin  9.78538  -  10. 

Z  log  sin  9.78538  - 10  =  37°  35'  + 
9.78527  - 10 
11 

Since  a  difference  of  16  in  the  log  makes  a  difference  of  1'  (or  of  60") 
in  the  angle,  a  difference  of  11  in  the  log  makes  a  difference  of  -J-^-  of 
60",  or  41",  in  the  angle. 

.-.  antilog  sin  9.78538-10  =  37°  35'  41",  Ans. 

Ex.  2.    Find  antilog  cos  9.96623  -  10. 

antilog  cos  9.96623  -  10  =  22°  19'  - 
9.96619  - 10 

-  of  60"  =  48" 
o 

antilog  cos  9.96623  -  10  =  22°  18'  12",  Ans. 

Ex.  3.    Find  antilog  cot  0.57603. 

antilog  cot  0.57603  =  14°  52'  - 
0.57601 

—  of  60"  =  2" 
51 

antilog  cot  0.57603  =  14°  51'  58",  Ans. 

Ex.  4.    Find  antilog  cos  9.60172  -  10. 

antilog  cos  9.60172  -  10  =  66°  27'- 
9.60157  - 10 

—  of  60"  =  31", 

antilog  cos  9.60172-10  =  66°  26'  29",  Ans. 


14  TRIGONOMETRY 

At  this  point  work  the  first  part  of  Exercise  15  of  Durell's 
Trigonometry. 


TABLE   IV.     AUXILIARY   FIVE  — PLACE   TABLE   FOR   SMALL   ANGLES 

(pp.  87-89) 

12.  The  Auxiliary  Table  of  Logarithms  of  Sine  and 
gent  for  Small  Angles  is  needed  because  when  an  an 
smaller  than  2°,  the  logarithms  of  the  sine  and  tangent  vary 
so  rapidly  that  ordinary  methods  of  interpolation  are  not 
sufficiently  accurate.  (The  same  is  true  for  the  cosine, 
cotangent,  and  tangent  when  the  angle  is  between  88°  and 
90°,  but  there  are  other  indirect  methods  of  meeting  such 
cases.) 

Table  IV  is  based  on  Art.  115  of  Plane  Trigonometry, 
where  it  is  shown  that  the  sine  (or  tangent)  of  a  small  angle 
is  approximately  the  same  in  value  as  the  number  of  radians 
in  the  angle.  Hence,  for  example,  to  find  sine  1°  21'  37", 
we  divide  the  number  of  seconds  in  1°  21'  37"  by  the  num- 
ber of  seconds  in  a  radian,  viz.  206,265.  This  process  is 
facilitated  by  Table  IV.  The  column  headed  "  in  this  table 
gives  the  number  of  seconds  in  each  angle  containing  an 
exact  number  of  minutes,  and  hence  is  an  aid  in  converting 
any  given  angle  into  seconds. 

In  the  column  headed  S'  is  given  the  log  of  206,265  (viz. 
5.31443),  modified  by  a  slight  correction  owing  to  the  change 
in  the  slight  differences  between  the  sine  of  a  small  angle 
and  the  radian  measure  of  that  angle.  Similarly  the  column 
headed  T'  gives  log  of  206/265  in  use  of  the  tangent.  (The 
columns  headed  S  and  T  give  the  cologs  corresponding  to 
the  S'  and  T'  columns.)  The  column  headed  log  sin  gives 
the  log  sin  or  final  answer  for  each  even  minute,  these  num- 
bers being  needed  also  in  guiding  the  work  in  the  inverse 
use  of  the  table.  Hence  — 


INTRODUCTION   TO   TABLES  15 

13.    To  find  the  log  sin  or  tangent  of  an  angle  less  than  2°. 

Find  the  number  of  seconds  in  the  given  angle  and  find  the 
log  of  this  number  in  Table  I ; 

Add  to  this  log  the  corresponding  log  in  column  S  or  T  ac- 
cording as  the  log  sin  or  log  tan  is  desired. 

T.  Find  log  sin  1°  26'  13". 
'  1°  26'  13"  =  5173" 

log  5173  =  3.71374 
S  (or  colog  206265)  =  4.68553  -  10 


.-.  log  1°  26'  13"  =  8.39927  -  10,  Ans. 

14.  To  find  the  angle  corresponding  to  a  given  log  sine  or 
log  tangent  (less  than  8.54282  -  10). 

Look  up  in  the  L.  Sin  column  the  number  nearest  in  size  to 
the  given  log;  and  set  down  the  number  on  the  same  row  with 
this  in  column  S'  or  T',  according  as  the  given  function  is  a 
sine  or  tangent; 

Add  the  given  log  function  to  the  number  set  down  from  the 
table; 

Find  the  antilog  of  the  result;  this  will  be  the  number  of 
seconds  in  the  required  angle. 

Ex.  Find  antilog  tan  8.39307. 

In  L.  Sin  column,  the  nearest  number  is  8.39310. 
Corresponding  to  this  is  T  =  5.31434 
Given  tan  =  8.39307 

antilog  13.70741  =  5098" 

=  1°  24'  58",  Ans. 

The  reason  for  the  above  process  is  seen  from  the  fact  that 

,  5098" 

rin  of  required  ^=206265^' 

.-.  206265  x  (sin  of  required  Z)  =  5098". 
.-.  log  206265  +  8.39307  =  log  5098", 


16  TRIGONOMETRY 

15.  Other  Uses  of  the  Auxiliary  Table  IV.     The  log  cosine 
of  an  angle  between  88°  and  90°  changes  so  rapidly  as  to 
make  direct  interpolation  inaccurate.     In  such  cases  use  the 
formula  cos  A  =  sin^  (990  _  A^ 

Thus,  for  example,  log  cos  &8°  47'  =  log  sin  1°  13',  and  the 
value  of  log  sin  1°  13'  can  be  obtained  by  Art.  14. 

The  log  cot  A,  when  A  is  between  88°  and  90°,  may 
tained  similarly. 

Also,  if  A  is  an  angle  between  88°  and  90°,  the  log  tan  A 
changes  so  rapidly  that  interpolation  is  inaccurate. 

In  this  case  use  tan  A  = 7. 

cot  A 

log  tan  A  =  colog  cot  A  =  colog  tan  (90°  —  A). 
Thus,  for  example,  log  tan  88°  47'  =  colog  tan  1°  13',  etc. 
At   this   point   work   the   first   part   of   Exercise    16    of 
Durell's  Trigonometry. 

TABLE  V.  FOUR-PLACE  TABLE  OF  THE  NATURAL  SINE,  COSINE, 
TANGENT,  AND  COTANGENT  FOR  EVERY  TEN  MINUTES  OF  THE 
QUADRANT  (pp.  91-96) 

16.  Method  of  using  Table  V. 

By  natural  trigonometric  functions  are  meant  the  actual 
numerical  (not  logarithmic)  values  of  these  functions.  Thus  ^ 
is  the  natural  sine  of  30°.  Interpolation  for  this  table  is 
made  in  the  same  general  way  as  for  Table  V^/'  \ 

Ex.  Find  natural  sine  27°  48'. 

N.  Sine  27°  40'  =  0.4643 

ft  of  26  = 21 

K  Sine  27°  48'  =  0.4664,  Ans. 

TABLE   VI.     FOUR-PLACE  TABLE   OF   LOGARITHMS  OF   NUMBERS 
1-2OOO    (pp.  97-1O1) 

17.  Method  of  using  Table  VI. 

In  using  the  four-place  log  of  a  number,  when  the  first  signifi- 
cant figure  of  the  number  is  1,  use  pp.  100-101 ;  otherwise  use 
pp.  98-99. 


INTRODUCTION   TO   TABLES  17 

In  finding  the  antilog  of  a  four-place  log,  if  the  given  log  is 
less  than  .3010,  use  pp.  100-101;  otherwise  use  pp.  98-99. 

At  this  point  work  the  latter  part  of  Exercises  3  and  4  of 
Durell's  Plane  Trigonometry. 

TABLE  VII.   POUR-PLACE  LOGARITHMIC  TABLE  OF  THE  TRIGONO- 

«3TRIC  FUNCTIONS  FOR  ANGLES  OF  THE  QUADRANT  EXPRESSED 
DECIMALLY   DIVIDED   DEGREES  (pp.  1O3-113) 
Method  of  using  Table  VII.     The  explanation  of  the 
methods  of  using  Table  III  given  in  Arts.  8—11  of  this  Intro- 
duction apply  in  general  to  the  use  of  Table  VII. 

Hence  we  need  only  illustrate  by  examples  the  application 
of  these  methods  to  the  table  in  hand. 

Ex.  1.    Find  log  sin  48.34°. 

log  sin  48.4°  =  9.8738  -  10  log  sin  48.3°  =  9.8731  -  10 

log  sin  48.3°  =  9.8731  -  10  T%  of  7  =  _       3 

7  log  sin  48.34°  =  9.8734  -  10,  Ans. 

Ex.  2.   Find  the  antilog  tan  0.2165. 

Z  log  tan  0.2165  =  58.7°+ 
2161 


Z  log  tan  0.2165  =  58.72°,  Ans. 

At  this  point  work  the  latter  part  of  Exercises  14  and  15 
of  Durell's  Trigonometry. 

19.  Four-place  Log  Functions  of  Angles  near  0°  or  90°.  As 
is  explained  in  Art.  12  of  this  Introduction,  when  an  angle  is 
less  than  2°,  the  logarithms  of  the  sine  and  tangent  vary 
so  rapidly  that  ordinary  methods  of  interpolation  are  not 
sufficiently  accurate.  To  get  an  accurate  log  function  in 
this  case  we  use  the  result  obtained  in  Art.  106  of  Plane 
Trigonometry,  viz  :  sine  or  tangent  of  a  very  small  Z  x 

-,.         -  Z  x  in  degrees 

=  no.  radians  in  Z  x,  or    =  -  0  -- 


18  TRIGONOMETRY 


.-.  log  sin  (or  tan)  of  small  Zx  =  log  x  +  colog  57.296 

=  log  x+  8.2419  -10. 

1  57.296° 

Also  when  x  is  small  cot  x  =  -  =  —  :  —      -- 

tan  x     x  in  degrees 

.-.  log  cot  small  Z  x=  1.7581  +  colog  x. 

Interpolation  '  also  is  not  accurate  for  log  cos,  log 
cot,  of  angles  between  88°  and  90°. 

When    A.    is  an  angle   between   88°  and  90°  proceec 
follows  : 

cos  A  =  sin  (90°  -A). 
/.  log  cos  A  =  log  sin  (90°  -  A)  =  8.2419  -  10  +  log  (90°  -A}. 

cot  A  =  tan  (90°  -A). 
.-.  log  cot  J.  =  log  tan  (90°  -J_)  =  8.2419  -10  +  log  (90°  -J.)- 

tan  A  =  —  3L       ...  log  tan  A  =  1.7581  -log  (90°-  A). 
cot  A 

Ex.  1.   Find  sin  0.876°. 

log  0.876°  =  9.9425  -  10 
colog  57.296°  =  8.2419  -  10 
.-.  log  sin  0.876°  =  8.1844  -  10,  Ans. 

Ex.  2.    Find  Z  log  sin  7.9592  -  10. 

17.9592  -  20 
8.2419  -  10 

antilog  9.7173  -  10  =  0.522°- 
.-.  Z  log  sin  7.9592  -  10  =  0.522°-,  Ans. 

At   this   point   work    the   latter  part  of  Exercise   16  of 
Durell's  Trigonometry. 

TABLE  VIII.     TABLE   FOR   CONVERTING   Mi  JUTES   AND   SECONDS 
INTO  THE  DECIMAL   PART  OF   A  DEGREE   (p.  114) 

20.    The  method  of  using  Table  VIII  is  evident  from  the 
form  of  the  table,  but  it  should  be  remembered  that  in  each 


INTRODUCTION   TO   TABLES  19 

decimal  equivalent  ending  in  a  significant  figure  the  last 
figure  is  supposed  to  repeat  indefinitely. 

Hence,  for  example,  we  have    36°  46'  =  36.7660+ 

=  36.77° 

Also    35°  43'         =  35.716° 
20"  =      .006° 


• 


.-.  35°  43'  20"  =  35.722 


=    35.72°,^s. 


TABLE  IX.     TABLE  FOB  CONVERTING   THE   DECIMAL   PARTS   OP   A 
DEGREE   INTO    MINUTES   AND   SECONDS  (p.  114) 

21.   The  method  of  using  Table  IX  is  also  evident  from  the 
table  itself. 


TABLE   I 

COMMON  LOGARITHMS 

OF  NUMBERS 
PAET  I 

)GARiTHMS   (WITH  CHARACTERISTICS)  OF  NUMBERS  1-100 


N. 

Log. 

N. 

Log. 

H. 

Log. 

N. 

Log. 

M 

—  Infinity 

30 

31 
32 
33 

1.47  712 

60 

61 
62 
63 

1.77  815 

90 

91 
92 
93 

1.95  424 

0.00  000 
0.30  103 
0.47  712 

1.49  136 
1.50  515 
1.51  851 

1.78  533 
1.79  239 
1.79  934 

1.95  904 
1.96  379 
1.96  848 

• 

0.60  206 
0.69-897 
0.77  815 

34 
35 
36 

1.53.  148 
l!f5"5630 

64 
65 
66 

1.80  618 
1.81  291 
1.81  954 

94 
95 
96 

1.97  313 
1.97  772 
1.98  227 

h 

9 
10 

Ls 

13 

0.84  510 
0.90  309 
0.95  424 

37 
38 
39 

40 

41 
42 
43 

1.56  820 
1.57  978 
1.59  106 

67 
68 
69 

70 

71 

72 
73 

1.82  607 
1.83  251 
1.83  885 

97 
98 
99 

100 

1.98  677 
1.99  123 
1.99  564 

1.00  000 

1.60  206 

1.84  510 

2.00  000 

1.04  139 
1.07  918 
1.11  394 

1.61  278 
1.62  325 
1.63  347 

1.85  126 
1.85  733 
1.86332 

.14 
15 
16 

1.14  613 
1.17  609 
1.20  412 

44 
45 
46 

1.64  345 
1.65  321 
1.66  276 

74 
75 
76 

1.86  923 
1.87  506 
1.88  081 

17 
18 
19 

20 

21 
22 
23 

1.23  045 

1.25  527 
1.27  875 

47 
48 
49 

50 

51 
52 
53 

1.67  210 
1.68  124 
1.69  020 

77 
78 
•  79 

80 

81 
82 
83 

1.88  649 
1.89  209 
1.89  763 

1.30  103 

1.69  897 

1.90  309 

1.32  222 
1.34  242 
1.36  173 

1.70  757 
1.71  600 
1.72  428 

1.90  849 
1.91  381 
1.91  908 

24 
25 
26 

1.38  021 
1.39  794 
1.41  497 

54 
55 
56 

1.73  239 
1.74  036 
1.74  819 

84 
85 
86 

1.92  428 
1.92  942 
1.93  450 

27 
28 
29 

30 

1.43  136 

1.44  716 
1.46  240 

57 
58 
59 

60 

1.75  587 
1.76  343 
1.77  085 

87 
88 
89 

90 

1.93  952 
1.94  448 
1.94  939 

1.47  712 

1.77  815 

1.95  424 

[21] 


PART  II 

MANTISSAS  OF  NUMBERS  1-10,000 

N. 

100 

01 
02 
03 

04 
05 
06 

07 
08 
09 
110 
11 
12 
13 

14 
15 
16 

17 
18 
19 
120- 
21 
22 
23 

24 
25 
26 

27 
28 
29 
130 

31 
32 
33 

34 
35 
36 

37 
38 
39 
140 
41 
42 
43 

44 
45 
46 

47 
48 
49 
150 

|     O 

00  000 

1 
043 

087 

JL 

130 

-L 

173 

5 

217 

260 

7 
303 

8 

346 

389  1 

432 
860 
01  284 

703 
02  119 
531 

938 
03  342 
743 

475 
903 
326 

745^ 
160 
572' 

979 
383 
782 

518 
945 
368 

787 
202 
612 

*019 
423 
822 

561 
988 
410 

828 
243. 
653 

*060 
463 
862 

604 
*030 
452 

870 
284 
694 

*100 
503 
902 

647 
*072 
494 

912 
325 
735 

*141 
543 
941 

689 
*115 
536 

953 

366 
776 

*181 
583 
981 

732 
*157 
.J78 

995 
407 
816 

*222 
623 
*021 

775 
*199 
620 

*036 
449 
857 

*262 
663 
*060 

817  1 
*242  1 
662 

*078 
490 
898 

*3^^J 

8c^ 
*269 
652 

*032 
408 
781 

*151 
518 
882 

04  139 

179 

218 

258 

297 

336 

376 

415 

454 

532 
922 
05  308 

690 
06  070 
446 

819 
07  188 
555 

571 
961 
<346 

729 
108 
483 

856 
225 
591 

610 
999 
385 

767 
145 
521 

893 

262 
628 

650 
*038 
423 

805 
183 
558 

930 

298 
664. 

689 
*077 
461 

843 
221 
595 

967 

335 
700 

727 
*115. 
500 

881 
258 
633 

*004 
372 
737 

766 
*154 
538. 

918 

296 
670 

*041 
408 
773 

805 
*192 
576 

956 
333 
707 

*078 
445 
809 

844 
*231 
614 

994 
371 
744 

*115 
482 
846 

918 

954 

990 

*027 

*063 

*099 

*135 

*171 

*207 

*243 

08  279 
636 
991 

09  342 
691 
10  037 

380 
721 
11  059 

314 
672 
*026 

377 
726 
072 

415 

755 
093 

350 
707 
*061 

412 
760 
106  . 

449 
789 
126 

386 
743 
*096 

447 
795 
140 

483 
823 
160 

422 
778 
*132 

482 
830 
175 

517 
857 
193 

458 
814 
*167 

517 
864 
209 

551 
890 
227 

493 
849 
*202 

552 
899 
243 

585 
924 
261 

529 
884 
*237 

587 
934 
278 

619 
958 
294 
~62~8~ 

565 
920 
*272 

621 
968 
312 

653 

992 
327 

600 
955 
*307 

656 
*003 
346 

687 
*025 
361 

394 

428 

461 

494 

528 

561 

594 

661 

694 

727 
12  057 
385 

710 
13  033 
354 

672 
988 
14  301 

760 
090 
418 

743 
066 
386 

704 
*019 
333 

793 
123 

450 

775 
098 
418 

735 
*051 
364 

826 
156 
483 

808 
130 
450 

767 
*082 
395 

860 
189 
516 

840 
162 
481 

799 
*114 
426 

893 
222 
548 

872 
194 
513 

830 
*145, 
457 

926 
254 
581 

905 
226 

545 

862 
*176 
489 

959 
287 
613 

937 
258 
577 

893 

*208 
520 

992 
320 
646 

969 
230 
609 

925 
*239 
551 

*024 
352 
678 

*001 
322 
640 

956 

*270 
582 

613 

644 

675 

706 

737 

768 

799 

829 

860 

891 

922 
15  229 
534 

836 
16  137' 
435 

732 
17  026 
319 

953 
259 
564 

866 
167 
465 

761 
056 
348 

983 
290 
594 

897 
197 
495 

791 
085 
377 

*014 
320 
625 

927 
227 
52-4 

820 
114 
406 

*045 
351' 
655 

957 
256 
554 

850 
143 
435 

*076 
381 
685 

987 
286 
584 

879 
173 
464 

*106 
412 
715 

*017 
316 
613 

909 
202 
493 

*137 
442 
746 

*047 
346 
643 

938 
231 
522 

*168 
473 
776 

*077 
376 
673 

967 
260 
551 

*198 

503 
806 

*107 
406  1 
702 

997  1 

289 
580  1 

609 

638 

667 

696 

725 

754 

782 

811 

840 

869  | 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

[22] 


.2-7 


[23] 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

200 

30  103 

125 

146 

168 

190 

••••••• 

211 

233 

255 

276 

298  1 

01 

320 

341 

363 

384 

406 

428 

449 

471 

492 

514 

02 

535 

557 

578 

600 

621 

643 

664 

685 

707 

728 

03 

750 

771 

792 

814 

835 

856 

878 

£29 

920 

942 

04 

963 

984 

*006 

*027 

*048 

*069 

*091 

*112 

*133 

*154 

05 

31  175 

197  • 

218 

239 

260 

281 

302 

323 

345 

366 

06 

387  . 

408 

429 

450 

471 

492 

513 

534 

555 

576 

07 

597 

618 

639 

660 

681 

702 

723 

744 

765 

785 

08 

806 

827 

848 

869 

890 

911 

931 

952 

973 

994 

09 

32  015 

035 

056 

077 

098 

118 

139 

160 

181 

201 

210 

222 

243 

263 

284 

305 

325 

346 

366 

387 

403 

11 

428 

449 

469 

490 

510 

531 

552 

572 

593 

61M 

12 

634 

654 

675 

695 

715 

736 

756 

777 

797 

13 

838 

858 

879 

899 

919 

940 

960 

980 

*001 

H 

14 

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072 

077 

082 

086 

091 

096 

73 

101 

106 

111 

116 

121 

126 

131 

136 

141 

146 

74 

151 

156 

161 

166 

171 

176 

181 

186 

191 

196 

75 

201 

206 

211 

216 

221 

226 

231 

236 

240 

245 

76 

250 

255 

260 

265 

270 

275 

280 

285 

290 

295 

77 

300 

305 

310 

315 

320 

325 

330 

335 

340 

345 

78 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

79 

399 

404 

409 

414 

419 

424 

429 

433 

438 

443 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

81 

498 

503 

507 

512 

517 

522 

527 

532 

537 

542 

82 

547 

552 

557 

562 

567 

571 

576 

581 

586 

591 

83 

596 

601 

606 

611 

616 

621 

626 

630 

635 

640 

•84 

645 

650 

655 

660 

665 

670 

675 

680 

685 

689 

85 

694 

699 

704 

709' 

714 

719 

724 

729 

734 

738 

86 

743 

748 

753 

758 

763 

768 

773 

778 

783 

787 

87 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 

88 

841 

846 

851 

856 

861 

866 

871 

876 

880 

885 

89 

890 

-895 

900 

905 

910 

915 

919 

924 

929 

934 

890 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

91 

988 

993 

998 

*002 

*007 

*012 

*017 

*022 

*027 

*032 

92 

95  036 

041 

046 

051 

056 

061 

066 

071 

075 

080 

93 

085 

090 

095 

100 

105 

109 

114 

119 

124 

129 

94 

134 

139 

143 

148 

153 

158 

163 

168 

173 

177 

95 

182 

187 

192 

197 

202 

207 

211 

216 

221 

226 

96 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 

97 

279 

284 

289 

294 

299 

303 

308 

313 

318 

323 

98 

328 

332 

337 

342 

347 

352 

357 

361 

366 

371 

99 

376 

381 

386 

390 

395 

400 

405 

410 

415 

419 

900 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

X. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

[37] 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

900 

95  424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

01 

472 

477 

482 

487 

492 

497 

501 

506 

511 

516 

02 

521 

525 

530 

535 

540 

545 

550 

554 

559 

564 

03 

569 

574 

578 

583 

588 

593 

598 

602 

607 

612 

04 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 

05 

665 

670 

674 

679 

684 

689 

694 

698 

703 

708 

06 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 

07 

761 

766 

770 

775 

780 

785 

789 

794 

799 

804 

08 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 

09 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

910 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

11 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 

12 

999 

*004 

*009 

*014 

*019 

*023 

*028 

*033 

*038 

*042 

13 

96  047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

14 

095 

099 

104 

109 

114 

118 

123 

128 

133 

137 

15 

142 

147 

152 

156 

161 

166 

171 

175 

180 

185 

16 

190 

194 

199 

204 

2d9 

2-13 

218 

223 

227 

232 

17 

237 

242 

246 

251 

256 

261 

265 

270 

275 

280 

18 

284 

289 

294 

298 

303 

308 

313 

317 

322 

327 

19 

332 

336 

341 

346 

350 

355 

360 

365 

369 

374 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

21 

426 

431 

435 

440 

445 

450 

454 

459 

464 

468 

22 

473 

478 

483 

487 

492 

497 

501 

506 

511 

515 

23 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 

24 

567 

572 

577 

581 

586 

591 

595 

600 

605 

609 

25 

614 

619 

624 

628 

633 

638 

642 

647 

652 

656 

26 

661 

666 

.670 

675 

680 

685 

689 

694 

699 

703 

27 

708 

713 

717 

722 

727 

731 

736 

741 

745 

750 

28 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 

29 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 

930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

31 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 

32 

942 

946 

951 

956 

960 

965 

970 

974 

979 

984 

33 

988 

993 

997 

*002 

*007 

*011 

*0}6 

*021 

*025 

*030 

34 

97  035 

039 

044 

049 

053 

058 

063 

067 

072 

077 

35 

081 

086 

090 

095 

100 

104 

109 

114 

118 

123 

36 

128 

132 

137 

142 

146 

151 

155 

160 

165 

169. 

37 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

38 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

39 

267 

271 

276 

280 

285 

290 

294 

2£9 

304 

308 

940 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 

41 

359 

364 

368 

373 

377 

382 

387 

391 

396 

400 

42 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 

43 

451 

456 

460 

465 

470 

474 

479 

483 

488 

493 

44 

497 

502 

506 

511 

516 

520 

525 

529 

534 

539 

45 

543 

548 

552 

557 

562 

566 

571 

575 

580 

585 

46 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 

47 

635 

640 

644 

649 

653 

658 

663 

667 

672 

676 

48 

681 

685 

690 

695 

699 

704 

708 

713 

717 

722 

49 

727 

731 

736 

740 

745 

749 

754 

759 

763 

768 

950 

772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

N. 

NMMB^H^B« 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

[38] 


ff. 

o 

1 

2 

3 

4 

5 

6 

7 

8 

9 

950 

97  772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

51 

818 

823 

827 

832 

836 

841 

845 

850 

855 

859 

52 

864 

868 

873 

877 

882 

886 

891 

896 

900 

905 

53 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 

lA 

955 

959 

964 

968 

973 

978 

982 

987 

991 

996 

1,5 

98  000 

005 

009 

014 

019 

023 

028 

032. 

037 

041 

£6 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 

57 

091 

096 

100 

105 

109 

114 

118 

123 

127 

132 

58 

137 

141 

146 

150 

155 

159 

164 

168 

173 

177 

59 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 

960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

61 

272 

277 

281 

286 

290 

295 

299 

304 

308 

313 

62 

318 

322 

'327 

331 

336 

340 

345 

349 

354 

358 

63 

363 

367 

372 

376 

381 

385 

390 

394 

29.9 

403 

64 

408 

412 

417 

421 

426 

430 

435 

439 

444 

448 

65 

453 

457 

462 

466 

471 

475 

480 

484 

489 

493 

66 

498 

502 

507 

511 

516 

520 

525 

529 

534 

538 

67 

543 

547 

552 

556 

561 

565 

570 

574 

579 

583 

68 

588 

592 

597 

601 

605 

610 

614 

619 

623 

628 

69 

632 

637 

641 

646 

650 

655 

659 

664 

668 

673 

970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

71 

722 

726 

731 

735 

740 

744 

749 

753 

758 

762 

72 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 

73 

811 

816 

820 

825 

829 

834 

838 

843 

847 

851 

74 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 

75 

900 

905 

909 

914 

918 

923 

927 

932 

936 

941 

76 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 

77 

989 

994 

998 

*003 

*007 

*012 

*016 

*021 

*025 

*029 

78 

99  034 

038 

043 

047 

052 

056 

061 

065 

069 

074 

79 

078 

083 

087 

092 

096 

100 

105 

109 

114 

118 

980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 

81 

167 

171 

176 

180 

185 

189 

193 

198 

202 

207 

82 

211 

216 

220 

224 

229  % 

233 

238 

242 

247 

251 

83 

255 

260 

264 

269 

273 

277 

282 

286 

291 

295 

84 

300 

304 

308 

313 

317 

322  . 

326 

330 

335 

339 

85 

344 

348 

352 

357 

361" 

366 

370 

374 

379 

383 

86 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

87 

432 

436 

441 

445 

'  449 

454 

458 

463 

467 

471 

88 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

89 

520 

524 

528 

533 

537 

542 

546 

550 

555 

559 

990 

564 

568 

572 

577 

581 

585 

590 

594 

599 

603 

91 

607 

612 

616 

621 

625 

629 

634 

638 

642 

647 

92 

651 

656 

660 

664 

669 

673 

677 

682 

686 

691 

93 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 

94 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 

95 

782 

787 

791 

795 

800 

804 

808 

813 

817 

822 

96 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 

97 

870 

874 

878 

883 

887 

891 

896 

900 

904 

909 

98 

913 

917 

922 

926 

930 

935 

939 

944 

948 

952 

99 

957 

961 

965 

970 

974 

978 

983 

987 

991 

996 

1000 

00  000 

004 

009 

013 

017 

022 

026 

030 

035 

039 

H. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

[39] 


TABLE   II 
LOGS  AND  COLOGS  OF   CERTAIN   MUCH-USED  NUMBEBS 


NUMBER 

LOGARITHM 

COLOGARITHM 

2 

0.3010300 

9.6989700-10 

3 

V2 
V3 

7T 

0.4771213 
0.1505150 
0.2385607 
0.4971499 

9.5228787-10 
9.8494850-10 
9.7614^93-10 
9.5028501-10 

7T2 

0.9942997 

9.0057003-10 

27T 

0.7981799 

9.2018201-10 

VTT 

0.2485749 

9.7514251-10 

57.2957795 

1.7581226 

8.2418774-10 

206264.806 

5.3144251 

4.6855749-10 

FIVE   PLACE 


2 

0.30103 

9.69897-10 

3 

0.47712 

9.52288-10 

V2 

0.15052 

9.84948-10 

V3 

0.23856 

9.76144-10 

IT 

0.49715 

9.50285-10 

7T2 

0.99430 

9.00570-10 

27T 

0.79818 

9.20182-10 

VTT 

0.24857 

9.75143-10 

57.2957795 

1.75812 

8.24188-10 

206264.806 

5.31443 

4.68557-10 

FOUR  PLACE 


2 

0.3010 

9.6990-10 

3 

0.4771 

9.5229-10 

V2 

0.1505 

9.8495-10 

V3 

0.2386 

9.7614-10 

7T 

0.4971 

9.5029-10 

7T2 

0.9943 

9.0057-10 

27T 

0.7982 

9.2018-10 

vV 

0.2486 

9.7514-10 

57.2956695 

1.7581 

8.2419-10 

206264.806 

5.3144 

4.6858-10 

[40] 


TABLE  III 


FIVE-PLACE  LOGARITHMS 


OF   THE 


SINE,   COSINE,   TANGENT,   AND 
COTANGENT 


FOR 


EACH  MINUTE  OF  THE  QUADRANT 


[41] 


0° 

/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

i 

2 
3 
4 

oo 
6.46  373 
6.76  476 
6.94  085 
7.06  579 

oo 
6.46  373 
6.76  476 
6.94  085 
7.06  579 

oo 
3.53  627 
3.23  524 
3.05  915 
2.93  421 

0.00  000 
0.00  000 
0.00  000 
0.00  000 
0.00  000 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

7.16  270 
7.24  188 
7.30  882 
7.36  682 
7.41  797 

7.16  270 
7.24  188 
7.30  882 
7.36  £82 
7.41  797 

2.83  730 
2.75  812 
2.69  118 
2.63  318 
2.58  203 

0.00  000 
0.00  000 
0.00  000 
0.00  000 
0.00  000 

55 
54 
53 
52 
51 

89° 

10 

11 
12 
13 
14 

7.46  373 
7.50  512 
7.54  291 
7.57  767 
7.60  985 

7.46  373 
7.50  512 
7.54  291 
7.57  767 
7.60  986 

2.53  627 
2.49  488 
2.45  709 
2.42  233 
2.39  014 

0.00  000 
0.00  000 
0.00  000 
0.00  000 
0.00  000 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

7.63  982 
7.66  784 
7.69  417 
7.71  900 
7.74  248 

7.63  982 
7.66  785 
7.69  418 
7.71  900 
7.74  248 

2.36  018 
2.33  215 
2.30  582 
2.28  100 
2.25  752 

0.00  000 
0.00  000 
9.99  999 
9.99  999 
9.99  999 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

7.76  475 
7.78  594 
7.80  615 
7.82  545 
7.84  393 

7.76  476 
7.78  595 
7.80  615 
7.82  546 
7.84  394 

2.23  524 
2.21  405 
2.19  385 
2.17  454 
2.15  606 

9.99  999 
9.99  999 
9.99  999 
9.99  999 
9.99  999 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

7.86  166 
7.87  870 
7.89  509 
7.91  088 
7.92  612 

7.86  167 
7.87  871 
7.89  510 
7.91  089 
7.92  613 

2.13  833 
2.12  129 
2.10  490 
2.08  911 
2.07  387 

9.99  999 
9.99  999 
9.99  999 
9.99  999 
9.99  998 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

7.94  084 
7.95  508 
7.96  887 
7.98  223 
7.99  520 

7.94  086 
7.95  510 
7.96  889 
7.98  225 
7.99  522 

2.05  914 
2.04  490 
2.03  111 
2.01  775 
2.00  478 

9.99  998 
9.99  998 
9.99  998 
9.99  998 
9.99  998 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

8.00  779 
8.02  002 
8.03  192 
8.04  350 
8.05  478 

8.00  781 
8.02  004 
8.03  194 
8.04  353 
8.05  481 

1.99  219 
1.97  996 
1.96  806 
1.95  647 
1.94  519 

9.99  998 
9.99  998 
9.99  997 
9.99  997 
9.99  997 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

8.06  578 
8.07  650 
8.08  696 
8.09  718 
8.10  717 

8.06  581 
8.07  653 
8.08  700 
8.09  722 
8.10  720 

1.93  419 
1.92  347 
1.91  300 
1.90  278 
1.89  280 

9.99  997 
9.99  997 
9.99  997" 
9.99  997 
9.99  996 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

8.11  693 
8.12  647 
8.13  581 
8.14  495 
8.15  391 

8.11  696 
8.12  651 
8.13  585 
8.14  500 
8.15  395 

1.88  304 
1.87  349 
1.86  415 
1.85  500 
1.84  605 

9.99  996 
9.99  996 
9.99  996 
9.99  996 
9.99  996 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

8.16  268 
8.17  128 
8.17  971 
8.18  798 
8.19  610 

8.16  273 
8.17  133 
8.17  976 
8.18  804 
8.19  616 

1.83  727 
1.82  867 
1.82  024 
1.81  196 
1.80  384 

9.99  995 
9.99  995 
9.99  995 
9.99  995 
9.99  995 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

8.20  407 
8.21  189 
8.21  958 
8.22  713 
8.23  456 

8.20  413 
8.21  195 
8.21  964 
8.22  720 
8.23  462 

1.79  587 
1.78  805  ' 
1.78  036 
1.77  280 
1.76  538 

9.99  994 
9.99  994 
9.99  994 
9.99  994 
9.99  994 

5 
4 
3 
2 
1 

60 

8.24  186 

8.24  192 

1.75  808 

9.99  993 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

r 

[42] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

1 

2 
3 

4 

8.24  186 
8.24  903 
8.25  609 
8.26  304 
8.26  988 

8.24  192 
8.24  910 
8.25  616 
8.26  312 
8.26  996 

1.75  808 
1.75  090 
1.74  384 
1.73  688 
1.73  004 

9.99  993 
9.99  993 
9.99  993 
9.99  993 
9.99  992 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

8.27  661 
8.28  324 
8.28  977 
8.29  621 
8.30  255 

8.27  669 
8.28  332 
8.28  986 
8.29  629 
8.30  263 

1.72  331 
1.71  668 
1.71  014 
1.70  371 
1.69  737 

9.99  992 
9.99  992 
9.99  992 
9.99  992 
9.99  991 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

8.30  879 
8.31  495 
8.32  103 
8.32  702 
8.33  292 

8.30  888 
8.31  505 
8.32  112 
8.32  711 
8.33  302 

1.69  112 
1.68  495 
1.67  888 
1.67  289 
1.66  698 

9.99  991 
9.99  991 
9.99  990 
9.99  990 
9.99  990 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

8.33  875 
8.34  450 
8.35  018 
8.35  578 
8.36  131 

8.33  886 
8.34  461 
8.35  029 
8.35  590 
8.36  143 

1.66  114 
1.65  539 
1.64  971 
1.64  410 
1.63  857 

9.99  990 
9.99  989 
9.99  989 
9.99  98,9 
'  9.99  989 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

8.36  678 
8.37  217 
8.37  750 
8.38  276 
8.38  796 

8.36  689 
8.37  229 
8.37  762 
8.38  289 
8.38  809 

1.63  311 
1.62  771 
1.62  238 
1.61  711 
1.61  191 

9.99  988 
9.99  988 
9.99  988 
9.99  987 
9.99  987 

40 

39 
38 
37 
36 

1° 

25 
26 
27 
28 
29 

8.39  310 
8.39  818 
8.40  320 
8.40  816 
8.41  307 

8.39  323 
8.39  832 
8.40  334 
8.40  830 
8.41  321 

1.60  677 
1.60  168 
1.59  666 
1.59  170 
1.58  679 

9.99  987 
9.99  986 
9.99  986 
9.99  986 
9.99  985 

35 
34 
33 
32 
31 

88° 

30 

31 

32 
33 
34 

8.41  792 
8.42  272 
8.42  746 
8.43  216 
8.43  680 

8.41  807 
8.42  287 
8.42  762 
8.43  232 
8.43  696 

1.58  193 
1.57  713 
1.57  238 
1.56  768 
1.56  304 

9.99  985 
9.99  985 
9.99  984 
9.99  984 
2.99  984 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

8.44.139 
8.44  594 
8.45  044 
8.45  589 
8.45  930 

8.44  156 
8.44  611 
8.45  061 
8.45  507 
8.45  948 

1.55  844 
1.55  389 
1.54  939 
1.54  493 
1.54  052 

9.99  983 
9.99  983 
9.99  983 
9.99  982 
9.99  982 

25 

24 
23 
22 
21 

40 

41 
42 
43 
44 

8.46  366 
8.46  799 
8.47  226 
8.47  650 
8.48  069 

8.46  385 
8.46  817 
8.47  245 
8.47  669 
8.48  089 

1.53  615 
1.53  183 
1.52  755 
1.52  331 
1.51  911 

9.99  982 
9.99  981 
9.99  981 
9.99  981 
9.99  980 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

8.48  485 
8.48  896 
8.49  304 
8.49  708 
8.50  108 

8.48  505 
8.48  917 
8.49  325 
8.49  729 
8.50  130 

1.51  495 
1.51  083 
1.50  675 
1.50  271 
1.49  870 

9.99  980 
9.99  979 
9.99  979 
9.99  979 
9.99  978 

15 

14 
13 
12 
11 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 

8.50  504 
8.50  897 
8.51  287 
8.51  673 
8.52  055 

8.50  527 
8.50  920 
8.51  310 
8.51  696 
8.52  079 

1.49  473 
1.49  080 
1.48  690 
1.48  304 
1.47  921 

9.99  978 
9.99  977 
9.99  977 
9.99  977 
9.99  976 

10 

9 
8 
7 
6 

8.52  434 
8.52  810 
8.53  183 
8.53  552 
8.53  919 

8.52  459 
8.52  835 
8.53  208 
8.53  578 
8.53  945 

1.47  541 
1.47  165 
1.46  792 
1.46  422 
1.46  055 

9.99  976 
9.99  975 
9.99  975 
9.99  974 
9.99  974 

5 
4 
3 
2 
1 

60 

8.54  282 

8.54  308 

1.45  692 

9.99  974 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[48] 


i 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 
1 
2 
3 

4 

8.54  282 
8.54  642 
8.54  999 
8.55  354 
8.55  705 

8.54  308 
8.54  669 
8.55  027 
8.55  382 
8.55  734 

1.45  692 
1.45  331 
1.44  973 
1.44  618 
1.44  266 

9.99  974 
9.99  973 
9.99  973 
9.99  972 
9.99  972 

60 

59 
58 
57 
56 

87° 

5 
6 
7 
8 
9 

8.56  054 
8.56  400 
8.56  743 
8.57  084 
8.57421 

8.56  083 
8.56  429 
8.56  773 
8.57  114 
8.57  452 

1.43  917 
1.43  571 
1.43  227 
1.42  886 
1.42  548 

9.99  971 
9.99  971 
9.99  970 
9.99  970 
9.99  969 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

8.57  757 
8.58  089 
8.58  419 
8.58  747 
8.59  072 

8.57  788 
8.58  121 
8.58  451 
8.58  779 
8.59  105 

1.42  212 
1.41  879 
1.41  549 
1.41  221 
1.40  895 

9.99  969 
9.99  968 
9.99  968 
9.99  967 
9.99  967 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

8.59  395 
8.59  715 
8.60  033 
8.6Q  349 
8.60  662 

8.59  428 
8.59  749 
8.60  068 
8.60  384 
8.60  698 

1.40  572 
1.40  251 
1.39  932 
1.39  616 
1.39  302 

9.99  967 
9.99  966 
9.99  966 
9.99  965 
9.99  964 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

8.60  973 
8.61  282 
8.61  589 
8.61  894 
8.62  196 

8.61  009 
8.61  319 
8.61  626 
8.61  931 
8.62  234 

1.38  991 
1.38  681 
1.38  374 
1.38  069 
1.37  766 

9.99  964 
9.99  963 
9.99  963 
9.99  962 
9.99  962 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

8.62  497 
8.62  795 
8.63  091 
8.63  385 
8.63  678 

8.62  535 
8.62  834 
8.63  131 
8.63  426 
8.63  718 

1.37465 
1.37  166 
1.36  869 
1.36  574 
1.36  282 

9.99  961 
9.99  961 
9.99  960 
9.99"960 
9.99  959 

35 
34 
33 
32 
31 

au 

30 

31 
32 
33 
34 

8.63  968 
8.64  256 
8.64  543 
8.64  827 
8.65  110 

8.64  009 
8.64  298 
8.64  585 
8.64  870 
8.65  154 

1.35  991 
1.35  702 
1.35  415 
1.35  130 
1.34  846 

9.99  959 
9.99  958 
9.99  958 
9.99  957 
9.99  956 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

8.65  391 
8.65  670 
8.65  947 
8.66  223 
8.66  497 

865435 
8.65  715 
8.65  993 
8.66  269 
8.66  543 

1.34  565 
1.34  285 
1.34  007 
1.33  731 
1.33  457 

9.99  956 
9.99  955 
9.99  955 
9.99  954 
9.99  954 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

8.66  769 
8.67  039 
8.67  308 
8.67  575 
8.67  841 

8.66  816 
8.67  087 
8.67  356 
8.67  624 
8.67  890 

1.33  184 
1.32  913 
1.32  644 
1.32  376 
1.32  110 

9.99  953 
9.99  952 
9.99  952 
9.99  951 
9.99  951 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

8.68  104 
8.68  367 
8.68  627 
8.68  886  ' 
8.69  144 

8.68  154 
8.68  417 
8.68  678 
8.68  938 
8.69  196 

1.31  846 
1.31  583 
1.31  322 
1.31  062 
1.30  804 

9.99  950 
9.99  949 
9.99  949 
9.99  948 
9.99  948 

15 
14 
13 
12 
11 

50 

51 
52 
53 

54 

8.69  400 
8.69  654 
8.69  907 
8.70  159 
8.70  409 

8.69  453 
8.69  708 
8.69  962 
8.70  214 
8.70  465 

1.30  547 
1.30  292 
1.30  038 
1.29  786 
1.29  535 

9.99  947 
9.99  946 
9.99  946 
9.99  945 
9.99  944 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

8.70  658 
8.70  903 
8.71  151 
8.71  395 
8.71  638 

8.70  714 
8.70  962 
8.71  208 
8.71  453 
8.71  697 

1.29  286 
1.29  038 
1.28  792 
1.28  547 
1.28  303 

9.99  944 
9.99  943 
9.99  942 
9.99  942 
9.99  941 

5 
4 
3 
2 
1 

60 

8.71  880 

8.71  940 

1.28  060 

9.99  940 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[44] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

i 

2 
3 
4 

8.71  880 
8.72  120 
8.72  359 
8.72  597 
8.72  834 

8.71  940 
8.72  181 
8.72  420 
8.72  659 
8.72  896 

1.28  060 
1.27  819 
1.27  580 
1.27  341 
1.27  104 

9.99  940 
9.99  940 
9.99  939 
9.99  938 
9.99  938 

60 

59 
58 

57 
56 

5 
6 
7 
8 
9 

8.73  069 
8.73  303 
8.73  535 
8.73  767 
8.73  997 

8.73  132 
8.73  366 
8.73  600 
8.73  832 
8.74  063 

1.26  868 
1.26  634 
1.26  400 
1.26  168 
1.25  937 

9.99  937 
9.99  936 
9.99  936 
9.99  935 
9.99  934 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

8.74  226 
8.74  454 
8.74  680 
8.74  906 
8.75  130 

8.74  292 
8.74  521 
8.74  748 
8.74  974 
8.75  199 

1.25  708 
1.25  479 
1.25  252 
1.25  026 
1.24  801 

9.99  934 
9.99  933 
9.99  932 
9.99  932 
9.99  931 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

8.75  353 
8.75  575 
8.75  795 
8.76  015 
8.76  234 

8.75  423 
8.75  645 
8.75  867 
8.76  087 
8.76  306 

1.24  577 
1.24  355 
1.24  133 
.23  913 
.23  694 

9.99  930 
9.99  929 
9.99  929 
9.99  928 
9.99  927 

45 
44 
43 
42 
'41 

20 

21 
22 
23 
24 

8.76  451 
8.76  667 
8.76  883 
8.77  097 
8.77  310 

8.76  525 
8.76  742 
8.76  958 
8.77  173 
8.77  387 

.23  475 
.23  258 
.23  042 
.22  827 
.22  613 

9.99  926 
9.99  926 
9.99  925 
9.99  924 
9.99  923 

40 

39 
38 
37 
36 

3° 

25 
26 
27 
28 
29 

8.77  522 
8.77  733 
8.77  943 
8.78  152 
8.78  360 

8.77  600 
8.77  811 
8.78  022 
8.78  232 
8.78  441 

1.22  400 
1.22  189 
1.21  978 
1.21  768 
1.21  559 

9.99  923 
9.99  922 
9.99  921 
9.99  920 
9.99  920 

35 
34 
33 
32 
31 

ftfi° 

30 

31 
32 
33 
34 

8.78  568 
8.78  774 
8.78  979 
8.79  183 
8.79  386 

8.78  649 
8.78  855 
8.79  061 
8.79  266 
8.79  470 

1.21  351 
1.21  145 
1.20  939 
1.20  734 
1.20  530 

9.99  919 
9.99  918 
9.99  917 
9.99  917 
9.99  916 

30 

29 
28 
27 
26 

ou 

35 
36 
37 
38 
39 

8.79  588 
8.79  789 
8.79  990 
8.80  189 
8.80  388 

8.79  673 
8.79  875 
8.80  076 
8.80  277 
8.80  476 

1.20  327 
1.20  125 
1.19  924 
1.19  723 
1.19  524 

9.99  915 
9.99  914 
9.99  913 
9.99  913 
9.99  912 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

8.80  585 
8.80  782 
8.80  978 
8.81  173 
8.81  367 

8.80  674 
8.80  872 
8.81  068 
8.81  264 
8.81  459 

1.19  326 
1.19  128 
1.18  932 
1.18  736 
1.18  541 

9.99  911 
9.99  910 
9.99  909 
9.99  909 
9.99  908 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

8.81  560 
8.81  752 
8.81  944 
8.82  134 
8.82  324 

8.81  653 
8.81  846 
8.82  038 
8.82  230 
8.82  420 

1.18  347 
1.18  154 
1.17  962 
1.17  770 
1.17  580 

9.99  907 
9.99  906 
9.99  905 
9.99  904 
9.99  904 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

8.82  513 
8.82  701 
8.82  888 
8.83  075 
8.83  261 

8.82  610 
8.82  799 
8.82  987 
8.83  175 
8.83  361 

1.17  390 
1.17  201 
1.17013 
1.16  825 
1.16  639 

9.99  903 
9.99  902 
9.99  901 
9.99  900 
9.99  899 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

8.83  446 
8.83  630 
8.83  813 
8.83  996 
8.84  177 

8.83  547 
8.83  732 
8.83  916 
8.84  100 
8.84  282 

1.16  453 
1.16  268 
1.16  084 
1.15  900 
1.15  718 

9.99  898 
9.99  898 
9.99  897 
9.99  896 
9.99  895 

5 
4 
3 
2 
1 

60 

8.84  358 

8.84  464 

1.15  536 

9.99  894 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[45] 


t 

L.  Sin. 

I.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

i 

2 
3 
4 

8.84  358 
8.84  539 
8.84  718 
8.84  897 
8.85  075 

8.84  464 
8.84  646 
8.84  826 
8.85  006 
8.85  185 

1.15  536 
1.15  354 
1.15  174 
1.14  994 
1.14  815 

9.99  894 
9.99  893 
9.99  892 
9.99  891 
9.99  891 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

8.85  252 
8.85  429 
8.85  605 
8.85  780 
8.85  955 

8.85  363 
8.85  540 
8.85  717 
8.85  893 
8.86  069 

1.14  637 
1.14  460 
1.14  283 
1.14  107 
1.13  931 

9.99  890 
9.99  889 
9.99  888 
9.99  887 
9.99  886 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

8.86  128 
8.86  301 
8.86  474 
8.86  645 
8.86  816 

8.86  243 
8.86  417 
8.86  591 
8.86  763 
8.86  935 

1.13  757 
1.13  583 
1.13  409 
1.13  237 
1.13  065 

9.99  885 
9.99  884 
9.99  883 
9.99  882 
9.99  881 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

8.86  987 
8.87  156 
8.87  325 
8.87  494 
8.87  661 

8.87  106 
8.87  277 
8.87  447 
8.87  616 
8.87  785 

1.12  894 
1.12  723 
1.12  553 
1.12  384 
1.12  215 

9.99  880 
9.99  879 
9.99  879 
9.99  878 
9.99  877 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

8.87  829 
8.87  995 
8.88  161 
8.88  326 
8.88  490 

8.87  953 
8.88  120 
8.88  287 
8.88  453 
8.88  618 

1.12  047 
1.11  880 
1.11  713 
1.11  547 
1.11  382 

9.99  876 
9.99  875 
9.99  874 
9.99  873 
9.99  872 

40 

39 
38 
37 
36 

85° 

4° 

25 
26 
27 
28 
29 

8.88  654 
8.88  817 
8.88  980 
8.89  142 
8.89  304 

8.88  783 
8.88  948 
8.89  111 
8.89  274 
8.89  437 

1.11  217 
1.11  052 
1.10  889 
1.10  726 
1.10  563 

9.99  871 
9.99  870 
9.99  869 
9.99  868 
9.99  867 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

8.89  464 
8.89  625 
8.89  784 
8.89  943 
8.90  102 

8.89  598 
8.89  760 
8.89  920 
8.90  080 
8.90  240 

1.10  402 
1.10  240 
1.10  080 
1.09  920 
1.09  760 

9.99  866 
9.99  865 
9.99  864 
9.99  863 
9.99  862 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

8.90  260 
8.90  417 
8.90  574 
8.90  730 
8.90  885 

8.90  399 
8.90  557 
8.90  715 
8.90  872 
8.91  029 

1.09  601 
1.09  443 
1.09  285 
1.09  128 
1.08  971 

9.99  861 
9.99  860 
9.99  859 
9.99  858 
9.99  857 

25 
24 
23 
22 
21 

40 

41- 
42 
43 
44 

8.91  040 
8.91  195 
8.91  349 
8.91  502 
8.91  655 

8.91  185 
8.91  340 
8.91  495 
8.91  650 
8.91  803 

1.08  815 
1.08  660 
1.08  505 
1.08  350 
1.08  197 

9.99  856 
9.99  855 
9.99  854 
9.99  853 
9.99  852 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

8.91  807 
8.91  959 
8.92  110 
8.92  261 
8.92  411 

8.91  957 
8.92  110 
8.92  262 
8.92  414 
8.92  565 

1.08  043 
1.07  890 
1.07  738 
1.07  586 
1.07  435 

9.99  851 
9.99  850 
9.99  848 
9.99  847 
9.99  846 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

8.92  561 
8.92  710 
8.92  859 
8.93  007 
8.93  154 

8.92  716 
8.92  866 
8.93  016 
8.93  165 
8.93  313 

1.07  284 
1.07  134 
1.06  984 
1.06  835 
1.06  687 

9.99  845 
9.99  844 
9.99  843 
9.99  842 
9.99  841 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

8.93  301 
8.93  448 
8.93  594 
8.93  740 
8.93  885 

8.93  462 
8.93  609 
8.93  756 
8.93  903 
8.94  049 

1.06  538 
1.06  391 
1.06  244 
1.06  097 
1.05  951 

9.99  840 
9.99  839 
9.99  838 
9.99  837 
9.99  836 

5 
4 
3 
2 
1 

60 

8.94.030 

8.94  195 

1.05  805 

9.99  834 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[46] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

1 

2 
3 
4 

8.94  030 
8.94  174 
8.94  317 
8.94  461 
8.94  603 

8.94  195 
8.94  340 
8.94  485 
8.94  630 
8.94  773 

1.05  805 
1.05  660 
1.05  515 
1.05  370 
1.05  227 

9.99  834 
9.99  833 
9.99  832 
9.99  831 
9.99  830 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

8.94  746 
8.94  887 
8.95  029 
8.95  170 
8.95  310 

8.94  917 
8.95  060 
8.95  202 
8.95  344 
8.95  486 

1.05  083 
1.04  940 
1.04  798 
1.04  656 
1.04  514 

9.99  829 
9.99  828 
9.99  827 
9.99  825 
9.99  824 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

8.95  450 
8.95  589 
8.95  728 
8.95  867 
8.96  005 

8.95  627 
8.95  767 
8.95  908 
8.96  047 
8.96  187 

1.04  373 
1.04  233 
1.04  092 
1.03  953 
1.03  813 

9.99  823 
9.99  822 
9.99  821 
9.99  820 
9.99  819 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

8.96  143 
8.96  280 
8.96  417 
8.96  553 
8.96  689 

8.96  325 
8.96  464 
8.96  602 
8.96  739 
8.96  877 

1.03  675 
1.03-  536 
1.03  398 
1.03  261 
1.03  123 

9.99  817 
9.99  816 
9.99  815 
9.99  814 
9.99  813 

45 

44 
43 
42 
41 

5° 

20 

21 
22 
23 
24 

8.96  825 
8.96  960 
8.97  095 
8.97  229 
8.97  363 

8.97  013 
8.97  150 
8.97  285 
8.97  421 
8.97  556 

1.02  987 
1.02  850 
1.02  715 
1.02  579 
1.02  444 

9.99  812 
9.99  810 
9.99  809 
9.99  808 
9.99  807 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

8.97  496 
8.97  629 
8.97  762 
8.97  894 
8.98  026 

8.97  691 
8.97  825 
8.97  959 
8.98  092 
8.98  225 

1.02  309 
1.02  175 
1.02  041 
1.01  908 
1.01  775 

9.99  806 
9.99  804 
9.99  803 
9.99  802 
9.99  801 

35 
34 
33 
32 
31 

84° 

30 

31 
32 
33 

34 

8.98  157 
8.98  288 
8.98  419 
8.98  549 
8.98  679 

8.98  358 
8.98  490 
8.98  622 
8.98  753 
8.98  884 

1.01  642 
1.01  510 
1.01  378 
1.01  247 
1.01  116 

9.99  800 
9.99  798 
9.99  797 
9.99  796 
9.99  795 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

8.98  808 
8.98  937 
8.99  066 
8.99  194 
8.99  322 

8.99  015 
8.99  145 
8.99  275 
8.99  405 
8.99  534 

1.00  985 
1.00  855 
1.00  725 
1.00  595 
1.00  466 

9.99  793 
9.99  792 
9.99  791 
9.99  790 
9.99  788 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

8.99  450 
8.99  577 
8.99  704 
8.99  830 
8.99  956 

8.99  662 
8.99  791 
8.99  919 
9.00  046 
9.00  174 

1.00  338 
1.00  209 
1.00  081 
0.99  954 
0.99  826 

9.99  787 
9.99  786 
9.99  785 
9.99  783 
9.99  782 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.00  082 
9.00  207 
9.00  332 
9.00  456 
9.00  581 

9.00  301 
9.00  427 
9.00  553 
9.00  679 
9.00  805 

0.99  699 
0.99  573 
0.99  447 
0.99  321 
0.99  195 

9.99  781 
9.99  780 
9.99  778 
9.99  777 
9.99  776 

15 

14 
13 
12 
11 

50 

51 
52 
53 
54 

9.00  704 
9.00  828 
9.00  951 
9.01  074 
9.01  196 

9.00  930 
9.01  055 
9.01  179 
9.01  303 
9.01  427 

0.99  070 
0.98  945 
0.98  821 
0.98  697 
0.98  573 

9.99  775 
9.99  773 
9.99  772 
9.99  771 
9.99  769 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.01  318 
9.01  440 
9.01  561 
9.01  682 
9.01  803 

9.01  550 
9.01  673 
9.01  796 
9.01  918 
9.02  040 

0.98  450 
0.98  327 
0.98  204 
0.98  082 
0.97  960 

9.99  768 
9.99  767 
9.99  765 
9.99  764 
9.99  763 

5 
4 
3 
2 
1 

60 

9.01  923 

9.02  162 

0.97  838 

9.99  761 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[47] 


1 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

1 
2 

3 
4 

9.01  923 
9.02  043 
9.02  163 
9.02  283 
9.02  402 

9.02  162 
9.02  283 
9.02  404 
9.02  525 
9.02  645 

0.97  838 
0.97  717 
0.97  596 
0.97  475 
0.97  355 

9.99  761 
9.99  760 
9.99  759 
9.99  757 
9.99  756 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.02  520 
9.02  639 
9.02  757 
9.02  874 
9.02  992 

9.02  766 
9.02  685 
9.03  005 
9.03  124 
9.03  242 

0.97  234 
0.97  115 
0.96  995 
0.96  876 
0.96  758 

9.99  755 
9.99  753 
9.99  752 
9.99  751 
9.99  749 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.03  109 
9.03  226 
9.03  342 
9.03  458 
9.03  574 

9.03  361 
9.03  479 
9.03  597 
9.03  714 
9.03  832 

0.96  639 
0.96  521 
0.96  403 
0.96  286 
0.96  168 

9.99  748 
9.99  747 
9.99  745 
9.99  744 
9.99  742 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.03  690 
9.03  805 
9.03  920 
9.04  034 
9.04  149 

9.03  948 
9.04  065 
9.04  181 
9.04  297 
9.04  413 

0.96  052 
0.95  935 
0.95  819 
0.95  703 
0.95  587 

9.99  741 
9.99  740 
9.99  738 
9.99  737 
9.99  736 

45 
44 
43 
42 
41 

fi° 

20 

21 
22 
23 
24 

9.04  262 
9.04  376 
9.04  490 
9.04  603 
9.04  715 

9.04  528 
9.04  643 
9.04  758 
9.04  873 
9.04  987 

0.95  472 
0.95  357 
0.95  242 
0.95  127 
0.95  013 

9.99  734 
9.99  733 
9.99  731 
9.99  730 
9.99  728 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

9.04  828 
9.04  940 
9.05  052 
9.05  164 
9.05  275 

9.05  101 
9.05  214 
9.05  328 
9.05  441 
9.05  553 

0.94  899 
0.94  786 
0.94  672 
0.94  559 
0.94  447 

9.99  727 
9.99  726 
9.99  724 
9.99  723 
9.99  721 

35 
34 
33 
32 
31 

83° 

u 

30 

31 
32 
33 
34 

9.05  386 
9.05  497 
9.05  607 
9.05  717 
9.05  827 

9.05  666 
9.05  778 
9.05  890 
9.06  002 
9.06  113 

0.94  334 
0.94  222 
0.94  110 
0.93  998 
0.93  887 

9.99  720 
9.99  718 
9.99  717 
9.99  716 
9.99  714 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.05  937 
9.06  046 
9.06  155 
9.06  264 
9.06  372 

9.06  224 
9.06  335 
9.06  445 
9.06  556 
9.06  666 

0.93  776 
0.93  665 
0.93  555 
0.93  444 
0.93  334 

9.99  713 
9.99  711 
9.99  710 
9.99  708 
9.99  707 

25 

24 
23 
22 
21 

40 

41 
42 
43 
44 

9.06  481 
9.06  589 
9.06  696 
9.06  804 
9.06  911 

9.06  775 
9.06  885 
9.06  994 
9.07  103 
9.07  211 

0.93  225 
0.93  115 
0.93  006 
0.92  897 
0.92  789 

9.99  705 
9.99  704 
9.99  702 
9.99  701 
9.99  699 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.07  018 
9.07  124 
9.07  231 
9.07  337 
9.07  442 

9.07  320 
9.07  428 
9.07  536 
9.07  643 
9.07  751 

0.92  680 
0.92  572 
0.92  464 
0.92  357 
0.92  249 

9.99  698 
9.99  696 
9.99  695 
9.99  993 
9.99  692 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.07  548 
9.07  653 
9.07  758 
9.07  863 
9.07  968 

9.07  858 
9.07  964 
9.08  071 
9.08  177 
9.08  283 

0.92  142 
0.92  036 
.  0.91  929 
0.91  823 
0.91  717 

9.99  690 
9.99  689 
9.99  687 
9.99  686 
9.99  684 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.08  072 
9.08  176 
9.08  280 
9.08  383 
9.08  486 

9.08  389 
9.08  495 
9.08  600 
9.08  705 
9.08  810 

0.91  611 
0.91  505 
0.91  400* 
0.91  295 
0.91  190 

9.99  683 
9.99  681 
9.99  680 
9.99  678 
9.99  677 

5 
4 
3 
2 
1 

60 

9.08  589 

9.08  914 

0.91  086 

9.99  675 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[48] 


t 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

82° 

« 

0 

1 

2 
3 
4 

9.08  589 
9.08  692 
9.08  795 
9.08  897 
9.08  999 

9.08  914 
9.09  019 
9.09  123 
9.09  227 
9.09  330 

0.91  086 
0.90  981 
0.90  877 
0.90  773 
0.90  670 

9.99  675 
9.99  674 
9.99  672 
9.99  670 
9.99  669 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.09  101 
9.09  202 
9.09  304 
9.09  405 
9.09  506 

9.09  434 
9.09  537 
9.09  640 
9.09  742 
9.09  845 

0.90  566 
0.90  463 
0.90  360 
0.90  258 
0.90  155 

9.99  667 
9.99  666 
9.99  664 
9.99  663 
9.99  661 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.09  606 
9.09  707 
9.09  807 
9.09  907 
9.10  006 

9.09  947 
9.10  049 
9.10  150 
9.10  252 
9.10  353 

0.90  053 
0.89  951 
0.89  853 
0.89  748 
0.89  647 

9.99  659 
9.99  658 
9.99  656 
9.99  655 
9.99  653 

50 

49 
48 
47 
46 

T 

15 
16 
17 
18 
19 

9.10  106 
9.10  205 
9.10  304 
9.10  402 
9.10  501 

9.10  454 
9.10  555 
9.1,0  656 
9.10  756 
9.10  856 

0.89  546 
0.89  445  • 
0.89  344 
0.89  244 
0.89  144 

9.99  651 
9.99  650 
9.99  648 
9.99  647 
9.99  645 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.10  599 
9.10  697 
9.10  795 
9.10  893 
9.10  990 

9.10  956 
9.11  056 
9.11  155 
9.11  254 
9.11  353 

0.89  044 
0.88  944 
0.88  845 
0.88  746 
0.88  647 

9.99  643 
9.99  642 
9.99  640 
9.99  638 
9.99  637 

40 

39 
38 
37 
33 

25 

26 
27 
28 
29 

9.11  087 
9.11  184 
9.11  281 
9.11  377 
9.11  474 

9.11  452 
9.11  551 
9.11  649 
9.11  747 
9.11  845 

0.88  548 
0.88  449 
0.88  351 
0.88  253 
0.88  155 

9.99  635 
9.99  633 
9.99  632 
9.99  630 
9.99  629 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.11  570 
9.11  666 
9.11  761 
9.11  857 
9.11  952 

9.11  943 
9.12  040 
9.12  138 
9.12  235 
9.12  332 

0.88  057 
0.87  960 
0.87"  862 
0.87  765 
0.87  668 

9.99  627 
9.99  625 
9.99  624 
9.99  622 
9.99  620 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.12  047 
9.12  142 
9.12  236 
9.12  331 
9.12  425 

9.12  428 
9.12  525 
9.12  621 
9.12  717 
9.12  813 

0.87  572 
0.87  475 
0.87  379 
0.87  283 
0.87  187   • 

9.99  618 
9.99  617 
9.99  615 
9.99  613 
9.99  612 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.12  519 
9.12  612 
9.12  706 
9.12  799 
9.12  892 

9.12  909 
9.13  004 
9.13  099 
'9.13  194 
9.13  289 

0.87  091 
0.86  996 
0.86  901 
0.86  806 
0.85  711 

9.99  610 
9.99  608 
9.99  607 
9.99  605 
9.99  603 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.12  985 
9.13  078 
9.13  171 
9.13  263 
9.13  355 

9.13  384 
9.13  478 
9.13  573 
9.13  667 
9.13  761 

0.86  616 
0.86  522 
0.86  427 
0.86  333 
0.86  239 

9.99  601 
9.99  600 
9.99  598 
9.99  596 
9.99  595 

15 

14 
13 
12 
11 

50 

51 
52 
53 
54 

9.13  447 
9.13  539 
9.13  630 
9.13  722 
9.13  813 

9.13  854 
9.13  948 
9.14  041 
9.14  134 
9.14  227 

0.86  146 
0.86  052 
0.85  959 
0.85  866 
0.85  773 

9.99  593 
9.99  591 
9.99  589 
9.99  588 
9.99  586 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.13  904 
9.13  994 
9.14  085 
9.14  175 
9.14  266 

9.14  320 
9.14  412 
9.14  504 
9.14  597 
9.14  688 

0.85  680 
0.85  588 
0.85  496 
0.85  403 
0.85  312 

9.99  584 
9.99  582 
9.99  581 
9.99  579 
9.99  577 

5 
4 
3 
2 
1 

60 

9.14  356 

9.14  780 

0.85  220 

9.99  575 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

t 

[49] 


I 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 
1 
2 
3 

4 

9.14  356 
9.14  445 
9.14  535 
9.14  624 
9.14  714 

9.14  780 
9.14  872 
9.14  963 
9.15  054 
9.15  145 

0.85  220 
0.85  128 
0.85  037 
0.84  946 
0.84  855 

y.yy  b7b 

9.99  574 
9.99  572 
9.99  570 
9.99  568 

bO 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.14  803 
9.14  891 
9-14  980 
9.15  069 
9.15  157 

9.15  236 
9.15  327 
9.15  417 
9.15  508 
9.15  598 

0.84  764 
0.84,  673 
0.84  583 
0.84  492 
0.84  402 

9.99  566 
9.99  565 
9.99  563 
9.99  561 
9.99  559 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.15  245 
9.15  333 
9.15  421 
9.15  508 
9.15  596 

9.15  688 
9.15  777 
9.15  867 
9.15  956 
9.16  0*6 

0.84  312 
0.84  223 
0.84  133 
0.84  044 
0.83  954 

9.99  557 
9.99  556 
9.99  554 
9.99  552 
9.99  550 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.15  683 
9.15  770 
9.15  857 
9.15  944 
9.16  030 

9.16  135 
9.16  224 
9.16  312 
9.16  401 
9.16  489 

0.83  865 
0  83  776 
0.83  688 
0.83  599 
0.83  511 

9.99  548 
9.99  546 
9.99  545 
9.99  543 
9.99  541 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.16  116 
9.16  203 
9.16  289 
9.16  374 
9.16  460 

9.16  577 
9.16  665 
9.16  753 
9.16  841 
9.16  928 

0.83  423 
0.83  335 
0.83  247 
0.83  159 
0.83  072 

9.99  539 
9.99  537 
9.99  535 
9.99  533 
9.99  532 

40 

39 
38 
37 
36 

81° 

00 

25 
26 
27 
28 
29 

9.16  545 
9.16  631 
9.16  716 
9.16  801 
9.16  886 

9.17016 
9.17  103 
9.17  190 
9.17  277 
9.17  363 

0.82  984 
0.82  897 
0.82  810 
0.82  723 
0.82  637 

9.99  530 
9.99  528 
9.99  526 
9.99  524 
9.99  522 

35 
34 
33 
32 
31 

O 

30 

31 
32 
33 
34 

9.16  970 
9.17  055 
9.17  139 
9.17  223 
9.17  307 

9.17  450 
9.17  536 
9.17  622 
9.17  708 
9.17  794 

0.82  550 
0.82  464 
0.82  378 
0.82  292 
0.82  206 

9.99  520 
9.99  518 
9.99  517 
9.99  515 
9.99  513 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.17  391 
9.17  474 
9.17  558 
9.17  641 
9.17  724 

9.17  880 
9.17  965 
9.18  051 
9.18  136 
•  9.18  221 

0.82  120 
0.82  035 
0.81  949 
0.81  864 
0.81  779 

9.99  511 
9:99  509 
9.99  507 
9.99  505 
9.99  503 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.17  807 
9.17  890 
9.17  973 
9.18  055 
9.18  137 

9.18  306 
9.18  391 
9.18  475 
9.18  560 
9.18  644 

0.81  694 
•  0.81  609 
0.81  525 
0.81  440 
0.81  356 

9.99  501 
9.99  499 
9.99  497 
9.99  495 
9.99  494 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.18  220 
9.18  302 
9.18  383 
9.18  465 
9.18  547 

9.18  728 
9.18  812 
9.18  896 
9.18  979 
9.19  063 

0.81  272 
0.81  188 
0.81  104 
0.81  021 
0.80  937 

9.99  492 
9.99  490 
9.99  488 
9.99  486 
9.99  484 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.18  628 
9.18  709 
9.18  790 
9.18  871 
9.18  952 

9.19  146 
9.19  229 
9.19  312 
9.19  395 
9.19  478 

0.80  854 
0.80  771 
0.80  688 
0.80  605 
0.80  522 

9.99  482 
9.99  480 
9.99  478 
9.99  476 
9.99U474 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.19  033 
9.19  113 
9.19  193 
9.19  273 
9.19  353 

9.19  561 
9.19  643 
9.19  725 
9.19  807 
9.19  889 

0.80  439 
0.80  357 
0.80  275 
0.80  193 
0.80  111 

9.99  472 
9.99  470 
9.99  468 
9.99  466 
9.99  464 

5 
4 
3 
2 
1 

60 

9  19  433 

9.19  971 

0.80  029 

9.99  462 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[50] 


I 

L.  Sin. 

L.  Tan. 

L.  Cotg. 

L.  Cos. 

80° 

0 

1 
2 
3 

4 

9.19  433 
9.19  513 
9.19  592 
9.19  672 
9.19  751 

9.19  971 
9.20  053 
9.20  134 
9.20  216 
9.20  297 

0.80  029 
0.79  947 
0.79  866 
0.79  784 
0.79  703 

9.99  462 
9.99  460 
9.99  458 
9.99  456 
9.99  454 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.19  830 
9.19  909 
9.19  988 
9.20  067 
9.20  145 

9.20  378 
9.20  459 
9.20  540 
9.20  621 
9.20  701 

0.79  622 
0.79  541 
0.79  460 
0.79  379 
0.79  299 

9.99  452 
9.99  450 
9.99  448 
9.99  446 
9.99  444 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.20  223 
9.20  302 
9.20  380 
9.20  458 
9.20  535 

9.20  782 
9.20  862* 
9.20  942 
9.21  022 
9.21  102 

0.79  218 
0.79  138 
0.79  058 
0.78  978 
Oi78  898 

9.99  442 
9.99  440 
9.99  438 
9.99  436 
9.99  434 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.20  613 
9.20  691 
9.20  768 
9.20  845 
9.20  922 

9.21  182 
9.21  261 
9.21  341 
9.21  420 
9.21  499 

0.78  818 
0.78  739 
0.78  659 
0.78  580 
0.78  501 

9.99  432 
9.99  429 
9.99  427 
9.99  425 
9.99  423 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.20  999 
9.21  076 
9.21  153 
9.21  229 
9.21  306 

9.21  578 
9.21  657 
9.21  736 
9.21  814 
9.21  893 

0.78  422 
0.78  343 
0.78  264 
0.78  186 
0.78  107 

9.99  421 
9.99  419 
9.99  417 
9.99  415 
9.99  413 

40 

39 
38 
37 
36 

9° 

25 
26 
27 
28 
29 

9.21  382 
9.21  458 
9.21  534 
9.21  610 
9.21  685 

9.21  971 
9.22  049 
9.22  127 
9:22  205 
9.22  283 

0.78  029 
0.77  951 
0.77  873 
0.77  795 
0.77  717 

9.99  411 
9.99  409 
9.99  407 
9.99  404 
9.99  402 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.21  761 
9.21  836 
9.21  912 
9.21  987 
9.22  062 

9.22  361 
9.22  438 
9.22  516 
9.22  593 
9.22  670 

0.77  639 
0.77  562 
0.77  484 
0.77  407 
0.77  330 

9.99  400 
9.99  39a 
9.99  396 
9.99  394 
9.99  392 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.22  137 
9.22  211 
9.22  286 
9.22  361 
9.22  435 

9.22  747 
9.22  824 
9.22  901 
9.22  977 
9.23  054 

0.77  253 
0.77  176 
0.77  099 
0.77  023 
0.76  946 

9.99  390 
9.99  388 
9.99  385 
9.99  383 
9.99  381 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.22  509 
9.22  583 
9.22  657 
9.22  731 
9.22  805 

9.23  130 
9.23  206 
9.23  283 
9.23  359 
9.23  435 

0.76  870 
0.76  794 
0.76  717 
0.76  641 
0.76  565 

9.99  379 
9.99  377 
9.99  375 
9.99  372 
9.99  370 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.22  878 
9.22  952 
9.23  025 
9.23  098 
9.23  171 

9.23  510 
9.23  586 
9.23  661 
9.23  737 
9  23  812 

0.76  490 
0.76  414 
0.76  339 
0.76  263 
0.76  188 

9.99  368 
9.99  366 
9.99  364 
9.99  362 
9.99  359 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.23  244 
9.23  317 
9.23  390 
9.23  462 
9.23  535 

9.23  887 
9.23  962 
9.24  037 
9.24  112 
9.24  186 

0.76  113 
0.76  038 
0.75  963 
0.75  888 
0.75  814 

9.99  357 
9.99  355 
9.99  353 
9.99  351 
9.99  348 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.23  607 
9.23  679 
9.23  752 
9.23  823 
9.23  895 

9.24  261 
9.24  335 
9.24  410 
9.24  484 
9.24  558 

0.75  739 
0.75  665 
0.75  590 
0.75  516 
0.75  442 

9.99  346 
9.99  344 
9.99  342 
9.99  340 
9.99  337 

5 
4 
3 
2 
1 

bO 

9.23  967 

9.24  632 

0.75  368 

9.99  335 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

1 

[51] 


10 

/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

79° 

0 

i 

2 
3 

4 

9.23  967 
9.24  039 
9.24  110 
9.24  181 
9.24  253 

9.24  632 
9.24  706 
9.24  779 
9.24  853 
9.24  926 

0.75  368 
0.75  294 
0.75  221 
0.75  147 
0.75  074 

9.99  335 
9.99  333 
9.99  331 
9.99  328 
9.99  326 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.24  324 
9.24  395 
9.24  466 
9.24  536 
9.24  607 

9.25  000 
9.25  073 
9.25  146 
9.25  219 
9.25  292 

0.75  000 
0.74  927 
0.74  854 
0.74  781 
0.74  708 

9.99  324 
9.99  322 
9.99  319 
9.99  317 
9.99  315 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.24  677 
9.24  748 
9.24  818 
9.24  888 
9.24  958 

9.25  365 
9.25  437 
9.25  510 
9.25  582 
9.25  655 

0.74  635 
0  74  563 
0.74  490 
0.74  418 
0.74  345 

9.99  313 
9.99  310 
9.99  308 
9.99  306 
9.99  304 

60 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.25  028 
9.25  098 
9.25  168 
9.25  237 
9.25  307 

9.25  727 
9.25  799 
9.25  871 
9.25  943 
9.26  015 

0.74  273 
0.74  201 
0.74  129 
0.74  057 
0.73  985 

9.99  301 
9.99  299 
9.99  297 
9.99  294 
9.99  292 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.25  376 
9.25  445 
9.25  514 
9.25  583 
9.25  652 

9.26  086 
9.26  158 
9.26  229 
9.26  301 
9.26  372 

0.73  914 
0.73  842 
0.73  771 
0.73  699 
0.73  628 

9.99  290 
9.99  288 
9.99  285 
9.99  283 
9.99  281 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

9.25  721 
9.25  790 
9.25  858 
9.25  927 
9.25  995 

-9.26  443 
9.26  514 
9.26  585 
9.26  655 
9.26  726 

0.73  557 
0.73  486 
0.73  415 
0.73  345 
0.73  274 

9.99  278 
9.99  276 
9.99  274 
9.99  271 
9.99  269 

35 
34 
33 
32 
31 

•60 

31 
32 
33 
34 

9.26  063 
9.26  131 
9.26  199 
9.26  267 
9.26  335 

9.26  797 
9.26  867 
9.26  937 
9.27  008 
9.27  078 

0.73  203 
0.73  133 
0.73  063 
0.72  992 
0.72  922 

9.99  267 
9.99  264 
9.99  262 
9.99  260 
9.99  257 

30 

29 

28 
27 
26 

35 
36 
37 
38 
39 

9.26  403 
9.26  470 
9.26  538 
9.26  605 
9.26  672 

9.27  148  . 
9.27  218 
9.27  288 
9.27  357 
9.27  427 

0.72  852 
0.72  782 
0.72  712 
0.72  643 
0.72  573 

9.99  255 
9.99  252 
9.99  250 
9.99  248 
9.99  245 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.26  739 
9.26  806 
9.26  873 
9.26  940 
9.27  007 

9.27  496 
9.27  566 
9.27  635 
9.27  704 
9.27  773 

0.72  504 
0.72  434 
0.72  365 
0.72  296 
0.72  227 

9.99  243 
9.99  241 
9.99  238 
9.99  236 
9.99  233 

20 

19 
18  • 
17 
16 

45 
46 
47 
48 
49 

9.27  073 
9.27  140 
9.27  206 
9.27  273 
9.27  339 

9.27  842 
9.27  911 
9.27  980 
9.28  049 
9.28  117 

0.72  158 
0.72  089 
0.72  020 
0.71  951 
0.71  883 

9.99  231 
9.99  229 
9.99  226 
9.99  224 
9.99  221 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.27405 
9.27  471 
9.27  537 
9.27  602 
9.27  668 

9.28  186 
9.28  254 
9.28  323 
9.28  391 
9.28  459 

0.71  814 
0.71  746 
0.71  677 
0.71  609 
0.71  541 

9.99  219 
9.99  217 
9.99  214 
9.99  212 
9.99  209 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.27  734 
9.27  799 
9.27  864 
9.27  930 
9.27  995 

9.28  527 
9.28  595 
9.28  662 
9.28  730 
9.28  798 

0.71  473 
0.71  405 
0.71  338 
0.71  270 
0.71  202 

9.99  207 
9.99  204 
9.99  202 
9.99  200 
9.99  197 

5 
4 
3 
2 
1 

60 

9.28  060 

9.28  865 

0.71  135 

9.99  195 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Cos. 

1 

[52] 


1 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

78° 

0 

1 

2 
3 

4 

9.28  060 
9.28  125 
9.28  190 
9.28  254 
9.28  319 

9.28  865 
9.28  933 
9.29  000 
9.29  067 
9.29  134 

0.71  135 
0.71  067 
0.71  000 
0.70  933 
0.70  866 

9.99  195 
9.99  192 
9.99  190 
9.99  187 
9.99  185 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.28  384 
9.28  448 
9.28  512 
9.28  577 
9.28  641 

9.29  201 
9.29  268 
9.29  335 
9.29  402 
9.29  468 

0.70  799 
0.70  732 
0.70  665 
0.70  598 
0.70  532 

9.99  182 
9.99  180 
9.99  177 
9.99  175 
9.99  172 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.28  705 
9.28  769 
9.28  833 
9.28  896 
9.28  960 

9.29  535 
9.29  601 
9.29  668 
9.29  734 
9.29  800 

0.70  465 
0.70  399 
0.70  332 
0.70  266 
0.70  200 

9.99  170 
9.99  167 
9.99  165 
9.99  162 
9.99  160 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.29  024 
9.29  087 
9.29  150 
9.29  214 
9.29  277 

9.29  866 
9.29  932 
9.29  998 
9.30  064 
9.30  130 

0.70  134 
0.70  068 
0.70  002 
0.69  936 
0.69  870 

9.99  157 
9.99  155 
9.99  152 
9.99  150 
9.99  147 

45 
44 
43 
42 
41 

20 

21 
22 
23 

24 

9.29  340 
9.29  403 
9.29  466 
9.29  529 
9.29  591 

9.30  195 
9.30  261 
9.30  326 
9.30  391 
9.30  457 

0.69  805 
0.69  739 
0.69  674 
0.69  609 
0.69  543 

9.99  145 
9.99  142 
9.99  140 
9.99  137 
9.99  135 

40 

39 
38 
37 
36 

11° 

25 
26, 
27 
28 
29 

9.29  654 
9.29  716 
9.29  779 
9.29  841 
9.29  903 

9.30  522 
9.30  587 
9.30  652 
9.30  717 
9.30  782 

0.69  478 
0.69  413 
0.69  348 
0.69  283 
0.69  218 

9.99  132 
9.99  130 
9.99  127 
9.99  124 
9.99  122 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.29  966 
9.30  028 
9.30  090 
9.30  151 
9.30  213 

9.30  846 
9.30  911 
9.30  975 
9.31  040 
9.31  104 

0.69  154 
0.69  089 
0.69  025 
0.68  960 
0.68  896 

9.99  119 
9.99  117 
9.99  114 
9.99  112 
9.99  109 

30 

29 
28 

27 
26 

35 
36 
37 
38 
39 

9.30  275 
9.30  336 
9.30  398 
9.30  459 
9.30  521 

9.31  168 
9.31  233 
9.31  297 
9.31  361  ' 
9.31  425 

0.68  832 
0.68  767 
0.68  703 
0.68  639 
0.68  575 

9.99  106 
9.99  104 
9.99  101 
9.99  099 
9.99  096 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.30  582 
9.30  643 
9.30  704 
9.30  765 
9.30  826 

9.31  489 
9.31  552 
9.31  616 
9.31  679 
9.31  743 

0.68  511 
0.68  448 
0.68  384 
0.68  321 
0.68  257 

9.99  093 
9.99  091 
9.99  088 
9.99  086 
9.99  083 

20 

19 
18 
17 
16 

45 
46 
47 
48 

49 

9.30  887 
9.30  947 
9.31  008 
9.31  068 
9.31  129 

9.31  806 
9.31  870 
9.31  933 
9.31  996 
9.32  059 

0.68  194 
0.68  130 
0.68  067 
0.68  004 
0.67  941 

9.99  080 
9.99  078 
9.99  075 
9.99  072 
9.99  070 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.31  189 
9.31  250 
9.31  310 
9.31  370 
9.31  430 

9.32  122 
9.32  185 
9.32  248 
9.32  311 
9.32  373 

0.67  878 
0.67  815 
0.67  752 
0.67  689 
0.67  627 

9.99  067 
9.99  064 
9.99  062 
9.99  059 
9.99  056 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.31  490 
9.31  549 
9.31  609 
9.31  669 
9.31  728 

9.32  436 
9.32  498 
9.32  561 
9.32  623 
9.32  685 

0.67  564 
0.67  502 
0.67  439 
0.67  377 
0.67  315 

9.99  054 
9.99  051 
9.99  048 
9.99  046 
9.99  043 

5 
4 
3 
2 
1 

60 

9.31  788 

9.32  747 

0.67  253 

9  99  040 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[53] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

77° 

0 

i 

2 
3 

4 

9.31  788 
9.31  847 
9.31  907 
9.31  966 
9.32  025 

9.32  747 
9.32  810 
9.32  872 
9.32  933 
9.32  995 

0.67  253 
0.67  190 
0.67  128 
0.67  067 
0.67  005 

9.99  040 
9.99  038 
9.99  035 
9.99  032 
9.99  030 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.32  084 
9.32  143 
9.32  202 
9.o2  261 
9.32  319 

9.33  057 
9.33  119 
9.33  180 
9.33  242 
9.33  303 

0.66  943 
0.66  881 
0.66  820 
0.66  758 
0.66  697 

9.99  027 
9.99  024 
9.99  022 
9.99  019 
9.99  016 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.32  378 
9.32  437 
9.32  495 
9.32  553 
9.32  612 

9.33  365 
9.33  426 
9.33  487 
9.33  548 
9.33  609 

0.66  635 
0.66  574 
0.66  513 
0.66  452 
0.66  391 

9.99  013 
9.99  Oil 
9.99  008 
9.99  005 
9.99  002 

50 

49 
48 
47 
46 

15 

16 
17 
18 
19 

9.32  670 
9.32  728 
9.32  786 
9.32  844 
9.32  902 

9.33  670 
9.33  731 
9.33  792 
9.33  853 
9.33  913 

0.66  330 
0.66  269 
0.66  208 
0.66  147 
0.66  087 

9.99  000 
9.98  997 
9.98  994 
9.98  991 
9.98  989 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.32  960 
9.33  018 
9.33  075 
9.33  133 
9.33  190 

9.33  974 
9.34  034 
9.34  095 
9.34  155 
9.34  215 

0.66  026 
0.65  966 
0.65  905 
0.65  845 
0.65  785 

9.98  986 
9.98  983 
9.98  980 
9.98  978 
9.98  975 

40 

39 
38 
37 
36 

12° 

25 
26 
27 
28 
29 

9.33  248 
9.33  305 
9.33  362 
9.33  420 
9.33  477 

9.34  276 
9.34  336 
9.34  396 
9.34  456 
9.34  516 

0.65  724 
0.65  664 
0.65  604 
0.65  544 
0.65  484 

9.98  972 
9.98  969 
9.98  967 
9.98  964 
9.98  961 

35 

34 
33 
32 
31 

• 

30 

31 
32 
33 
34 

9.33  534 
9.33  591 
9.33  647 
9.33  704 
9.33  761 

9.34  576 
9.34  635 
9.34  695 
9.34  755 
9.34  814 

0.65  424 
0.65  365 
0.65  305 
0.65  245 
0.65  186 

9.98  958 
9.98  955 
9.98  953 
9.98  950 
9.98  947 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.33  818 
9.33  874 
9.33  931 
9.33  987 
9.34  043 

9.34  874 
9.34  933 
9.34  992 
9.35  051 
9.35  111 

0.65  126 
0.65  067 
0.65  008 
0.64  949 
0.64  889 

9.98  944 
9.98  941 
9.98  938 
9.98  936 
9.98  933 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.34  100 
9.34  156 
9.34  212 
9.34  268 
9.34  324 

9.35  170 
9.35  229 
9.35  288 
9.35  347 
9.35  405 

0.64  830 
0.64  771 
0.64  712 
0.64  653 
0.64  595 

9,98  930 
9.98  927 
9.98  924 
9.98  921 
9.98  919 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.34  380  ' 
9.34  436 
9.34  491 
9.34  547 
9.34  602 

9.35  464 
9.35  523 
9.35  581 
9.35  640 
9.35  698 

0.64  536 
0.64  477 
0.64  419 
0.64  360 
0.64  302 

9.98  916 
9.98  913 
9.98  910 
9.98  907 
9.98  904 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.34  658 
9.34  713 
9.34  769 
9.34  824 
9.34  879 

9.35  757 
9.35  815 
9.35  873 
9.35  931 
9.35  989 

0.64  243 
0.64  185 
0.64  127 
0.64  069 
0.64  Oil 

9.98  901 
9.98  898 
9.98  896 
9.98  893 
9.98  890 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.34  934 
9.34  989 
9.35  044 
9.35  099 
9.35  154 

9.36  047 
9.36  105 
9.36  163 
9.36  221 
9.36  279 

0.63  953 
0.63  895 
0.63  837 
0.63  779 
0.63  721 

9.98  887 
9.98  884 
9.98  881 
9.98  878 
9.98  875 

5 
4 
3 
2 
1 

60 

9.35  209 

9.36  336 

0.63  664 

9.98  872 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[54] 


1Q 

/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

76° 

0 

1 

2 
3 
4 

9.35  209 
9.35  263 
9.35  318 
9.35  373 
9.35  427 

9.36  336 
9.36  394 
9.36  452 
9.36  509 
9.36  566 

0.63  664 
0.63  606 
0.63  548 
0.63  491 
0.63  434 

9.98  872 
9.98  869 
9.98  867 
9.98  864 
9.98  861 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.35  481 
9.35  536 
9.35  590 
9.35  644 
9.35  698 

9.36  624 
9.36  681 
9.36  738 
9.36  795 
9.36  852 

0.63  376 
0.63  319 
0.63  262 
0.63  205 
0.63  148 

9.98  858 
9.98  855 
9.98  852 
9.98  849 
9.98  846 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.35  752 
9.35  806 
9.35  860 
9.35  914 
9.35  968 

9.36  909 
9.36  966 
9.37  023 
9.37  080 
9.37  137 

0.63  091 
0.63  034 
0.62  977 
0.62  920 
0.62  863 

9.98  843 
9.98  840 
9.98  837 
9.98  834 
9.98  831 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.36  022 
9.36  075 
9.36  129   . 
9.36  182 
9.36  236 

9.37  193 
9.37  250 
9.37  306 
9.37  363 
9.37419 

0.62  807 
0.62  750 
0.62  694 
0.62  637 
0.62  581 

9.98  828 
9.98  825 
9.98  822 
9.98  819 
9.98  816 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.36  289 
9.36  342 
9.36  395 
9.36  449 
9.36  502 

9.37  476 
9.37  532 
9.37  588 
9.37  644 
9.37  700 

0.62  524 
0.62  468 
0.62  412 
0.62  356 
0.62  300 

9.98  813 
9.98  810 
9.98  807 
9.98  804 
9.98  801 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

9.36  555 
9.36  608 
9.36  660 
9.36  713 
9.36  766 

9.37  756 
9.37  812 
9.37  868 
9.37  924 
9.37  980 

0.62  244 
0.62  188 
0.62  132 
0.62  076 
0.62  020 

9.98  798 
9.98  795 
9.98  792 
9.98  789 
9.98  786 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.36  819 
9.36  871 
9.36  924 
9.36  976 
9.37  028 

9.38  035 
9.38  091 
9.38  147 
9.38  202 
9.38  257 

0.61  965 
0.61  909 
0.61  853 
0.61  798 
0.61  743 

9.98  783 
9.98  780 
9.98  777 
9.98  774 
9.98  771 

30 

29 
28 
27 
26 

35 
36 
37 
38 

39 

9.37  081 
9.37  133 
9.37  185 
9.37  237 
9.37  289 

9.38  313 
9.38  368 
9.38  423 
9.38  479 
9.38  534 

0.61  687 
0.61  632 
0.61  577 
0.61  521 
0.61  466 

9.98  768 
9.98  765 
9.98  762 
9.98  759 
9.98  756 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.37  341 
9.37  393 
9.37  445 
9.37  497 
9.37  549 

9.38  589 
9.38  644 
9.38  699 
9.38  754 
9.38  808- 

0.61  411 
0.61  356 
0.61  301 
0.61  246 
0.61  192 

9.98  753 
9.98  750 
9.98  746 
9.98  743 
9.98  740 

20 

19 
18 
17 
16 

45 

46 
47 
48 
49 

9.37  600 
9.37  652 
9.37  703 
9.37  755 
9.37  806 

9.38  863 
9.38  918 
9.38  972 
9.39  027 
9.39  082 

0.61  137 
0.61  082 
0.61  028 
0.60  973 
0.60  918 

9.98  737 
9.98  734 
9.98  731 
9.98  728 
9.98  725 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.37  858 
9.37  909 
9.37  960 
9.38  Oil 
9.38  062 

9.39  136 
9.39  190 
9.39  245 
9.39  299  ' 
9.39  353 

0.60  864 
0.60  810 
0.60  755 
0.60  701 
0.60  647 

9.98  722 
9.98  719 
9.98  715 
9.98  712 
9.98  709 

10 

9 

8 
7 
6 

55 

56 
57 
58 
59 

9.38  113 
9.38  164 
9.38  215 
9.38  266 
9.38  317 

9.39  407 
9.39  461 
9.39  515 
9.39  569 
9.39  623 

0.60  593 
0.60  539 
0.60  485 
0.60  431 
0.60  377 

9.98  706 
9.98  703 
9.98  700 
9.98  697 
9.98  694 

5 
4 
3 
2 
1 

60 

9.38  368 

9.39  677 

0.60  323 

9  98  690 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

; 

[55] 


1 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

75C 

0 

1 

2 
3 

4 

9.38  3b8 
9.38  418 
9.38  469 
9.38  519 
9.38  570 

9.39  677 
'  9.39  731 
9.39  785 
9.39  838 
9.39  892 

0.60  323 
0.60  269 
0.60  215 
0.60  162 
0.60  108 

9.98  690 
9.98  687 
9.98  684 
9.98  681 
9.98  678 

bu 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.38  620 
9.38  670 
9.38  721 
9.38  771 
9.38  821 

9.39  945 
9.39  999 
9.40  052 
9.40  106 
9.40  159 

0.60  055 
0.60  001 
0.59  948 
0.59  894 
0.59  841 

9.98  675 
9.98  671 
9.98  668 
9.98  665 
9.98  662 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.38  871 
9.38  921 
9.38  971 
9.39  021 
9.39  071 

9.40  212 
9.40  266 
9.40  319 
9.40  372 
9.40  425 

0.59  788 
0.59  734 
0.59  681 
0.59  628 
0.59  575  • 

9.98  659 
9.98  656 
9.98  652 
9.98  649 
9.98  646 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.39  121 
9.39  170 
9.39  220 
9.39  270 
9.39  319 

9.40  478 
9.40  531 
9.40  584 
9.40  636 
9.40  689 

0.59  522 
0.59  469 
0.59  416 
0.59  364 
0.59  311 

9.98  643 
9.98  640 
9.98  636 
9.98  633 
9.98  630 

45 
44 
43 
42 
41 

Wc 

*0 
21 
22 

23 
24 

9.39  369 
9.39  418 
9.39  467 
9.39  517 
9.39  566 

9.40  742 
9.40  795 
9.40  847 
9.40  900 
9.40  952 

0.59  258 
0.59  205 
0.59  153 
0.59  100 
0.59  048 

9.98  627 
9.98  623 
9.98  620 
9.98  617 
9.98  614 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

9.39  615 
9.39  664 
9.39  713 
9.39  762 
9.39  811 

9.41  005 
9.41  057 
9.41  109 
9.41  161 
9.41  214 

0.58  995 
0.58  943 
0.58  891 
0.58  839 
0.58  786 

9.98  610 
9.98  607 
9.98  604 
9.98  601 
9.98  597 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.39  860 
9.39  909 
9.39  958 
9.40  006 
9.40  055 

9.41  266 
9.41  318 
9.41  370 
9.41  422 
9.41  474 

0.58  734 
0.58  682 
0.58  630 
0.58  578 
0.58  526 

9.98  594 
9.98  591 
9.98  588 
9.98  584 
9.98  581 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.40  103 
9.40  152 
9.40  200 
9.40  249 
9.40  297 

9.41  526 
9.41  578 
9.41  629 
9.41  681 
9.41  733 

0.58  474 
0.58  422 
0.58  371 
0.58  319 
0.58  267 

9.98  578 
9.98  574 
9.98  571 
9.98  568 
9.98  565 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.40  346 
9.40  394 
9.40  442 
9.40  490 
9.40  538 

9.41  784 
9.41  836 
9.41  887 
9.41  939 
9.41  990 

0.58  216 
0.58  164 
0.58  113 
0.58  061 
.0.58  010 

9.98  561 
9.98  558 
9.98  555 
9.98  551 
9.98  548 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.40  586 
9.40  634 
9.40  682 
9.40  730 
9.40  778 

9.42  041 
9.42  093 
9.42  144 
9.42  195 
9.42  246 

0.57  959 
0.57  907 
0.57  856 
0.57  805 
0.57  754 

9.98  545 
9.98  541 
9.98  538 
9.98  535 
9.98  531 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.40  825 
9.40  873 
9.40  921 
9.40  968 
9.41  016 

9.42  297 
9.42  348 
9.42  399 
9.42  450 
9.42  501 

0.57  703 
0.57  652 
0.57  601 
C.57  550 
0.57  499 

9.98  528 
9.98  525 
9.98  521 
9.98  518 
9.98  515 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.41  063 
9.41  111 
9.41  158 
9.41  205 
9.41  252 

9.42  552 
9,42  603 
9.42  653  ^ 
9.42  704 
9.42  755 

0.57  448 
0.57  397 
0.57  347 
0.57  296 
0.57  245 

9.98  511 
9.98  508 
9.98  505 
9.98  501 
9.98  498 

5 
4 
3 
2 
1 

60 

9.41  300 

9.42  805 

0.57  195 

9.98  494 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

I 

[56] 


i 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

74C 

0 

i 

2 
3 
4 

9.41  300 
9.41  347 
9.41-394 
9.41  441 
9.41  488 

9.42  805 
9.42  856 
9.42  906 
9.42  957 
9.43  007 

0.57  195 
0.57  144 
0.57  094 
0.57  043 
0.56  993 

9.98  494 
9.98  491 
9.98  488 
9.98  484 
9.98  481 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.41  535 
9.41  582 
9.41  628 
9.41  675 
9.41  722 

9.43  057 
9.43  108 
9.43  158 
9.43  208 
9.43  258 

0.56  943 
0.56  892 
0.56  842 
0.56  792 
0.56  742 

9.98  477 
9.98  474 
9.98  471 
9.98  467 
9.98  464 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.41  768 
9.41  815 
9.41  861 
9.41  908 
9.41  954 

9.43  308 
9.43  358 
9.43  408 
9.43  458 
9.43  508 

0.56  692 
0.56  642 
0.56  592 
0.56  542 
0.56  492 

9.98  460 
9.98  457 
9.98  453 
9.98  450 
9.98  447 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.42  001 
9.42  047 
9.42  093 
9.42  140 
9.42  186 

9.43  558 
9.43  607 
9.43  657 
9.43  707 
9.43  756 

0.56  442 
0.56  393 
0.56  343 
0.56  293 
0.56  244 

9.98  443 
9.98  440 
9.98  436 
9.98  433 
9.98  429 

45 
44 
43 
42 
41 

20 

21 
22 
23 

24 

9.42  232 
9.42  278 
9.42  324 
9.42  370 
9.42  416 

9.43  806 
9.43  855 
9.43  905 
9.43  954 
9.44  004 

0.56  194 
0.56  145 
0.56  095 
0.56  046 
0.55  996 

9.98  426 
9.98  422 
9.98  419 
9.98  415 
9.98  412 

40 

39 
38 
37 
36 

15° 

25 
26 
27 
28 
29 

9.42  461 
9.42  507 
9.42  553 
9.42  599 
9.42  644 

9.44  053 
9.44  102 
9.44  151 
9.44  201 
9.44  250 

0.55  947 
0.55  898 
0.55  849 
0.55  799 
0.55  750 

9.98  409 
9.98  405 
9.98  402 
9.98  398 
9.98  395 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.42  690 
9.42  735 
9.42  781 
9.42  826 
9.42  872 

9.44  299 
9.44  348 
9.44  397 
9.44  446 
9.44  495 

0.55  701 
0.55  652 
0.55  603 
0.55  554 
0.55  505 

9.98  391 
9.98  388 
9.98  384 
9.98  381 
9.98  377 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.42  917 
9.42  962 
9.43  008 
9.43  053 
9.43  098 

9.44  544 
9.44  592 
9.44  641 
9.44  690 
9.44  738 

0.55  456 
0.55  408 
0.55  359 
0.55  310 
0.55  262 

9.98  373 
9.98  370 
9.98  366 
9.98  363 
9.98  359 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.43  143 
9.43  188 
9.43  233 
9.43  278 
9.43  323 

9:44  787 
9.44  836 
9.44  884 
9.44  933 
9.44  981 

0.55  213 
0.55  164 
0.55  116 
0.55  067 
0.55  019 

9.98  356 
9.98  352 
9.98  349 
9.98  345 
9.98  342 

20 

19 
18 
17 
16 

45 
46 
47 
48 

49 

9.43  367 
9.43  412 
9.43  457 
9.43  502 
9.43  546 

9.45  029 
9.45  078 
9.45  126 
9.45  174 
9.45  222 

0.54  971 
0.54  922 
0.54  874 
0.54  826 
0.54  778 

9.98  338 
9.98  334 
9.98  331 
9.98  327 
9.98  324 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.43  591 
9.43  635 
9.43  680 
9.43  724 
9.43  769 

9.45  271 
945  319 
9.45  367 
9:45  414 
9.45  463 

0.54  729 
0.54  681 
0.54  633 
0.54  585 
0.54  537 

9.98  320 
9.98  317 
9.98  313 
9.98  309 
9.98  306 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.43  813 
9.43  857 
9.43  901 
9.43  946 
9.43  990 

9.45  511 
9.45  559 
9-45  606 
9.45  654 
9.45  702 

0.54  489 
0.54  441 
0.54  394 
0.54  346 
0.54  298 

9.98  302 
9.98  299 
9.98  295 
9.98  291 
9.98  288 

5 
4 
3 
2 
1 

60 

9.44  034 

9.45  750 

0.54  250 

9.98  284 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[57] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

73° 

0 

1 

2 
3 
4 

9.44  034 
9.44  078 
9.44  122 
9.44  166 
9.44  210 

9.45  750 
9.45  797 
9.45  845 
9.45  892 
9.45  940 

0.54  250 
0.54  203 
0.54  155 
0.54  108 
0.54  060 

9.98  284 
9.98  281 
9.98  277 
9.98  273 
9.98  270 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.44  263 
9.44  297 
9.44  341 
9.44  385 
9.44  428 

9.45  987 
9.46  035 
9.46  082 
9.46  130 
9.46  177 

0.54  013 
0.53  965 
0.53  918 
0.53  870 
0.53  823 

9.98  266 
9.98  262 
9.98  259 
9.98  255 
9.98  251 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.44  472 
9.44  516 
9.44  559 
9.44  602 
9.44  646 

9.46  224 
9.46  271 
9.46  319 
9.46  366 
9.46  413 

0.53  776 
0.53  729 
0.53  681 
0.53  634 
0.53  587 

9.98  248 
9.98  244 
9.98  240 
9.98  237 
9.98  233 

60 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.44  689 
9.44  733 
9.44  776 
9.44  819 
9.44  862 

9.46  460 
9.46  507 
9.46  554 
9.46  601 
9.46  648 

0.53  540 
0.53  493 
0.53  446 
0.53  399 
0.53  352 

9.98  229 
9.98  226 
9.98  222 
9.98  218 
9.98  215 

45 

44 
43 
42 
41 

20 

21 
22 
23  . 
24 

9.44  905 
9.44  948 
9.44  992 
9.45  035 
9.45  077 

9.46  694 
9.46  741 
9.46  788 
9.46  835 
9.46  881 

0.53  306 
0.53  259 
0.53  212 
0.53  165 
0.53  119 

9.98  211 
9.98  207 
9.98  204 
9.98  200 
9.98  196 

40 

39 
38 
37 
36 

1fic 

25 
26 
27 
28 
29 

9.45  120 
9.45  163 
9.45  206 
9.45  249 
9.45  292 

9.46  928 
9.46  975 
9.47  021 
9.47  068 
9.47  114 

0.53  072 
0.53  025 
0.52  979 
0.52  932 
0.52  886 

9.98  192 
9.98  189 
9.98  185 
9.98  181 
9.98  177 

35 
34 
33 
32 
31 

AvF 

30 

31 
32 
33 
34 

9.45  334 
9.45  377 
9.45  419 
9.45  462 
9.45  504 

9.47  160 
9.47  207 
9.47  253 
9.47  299 
9.47  346 

0.52  840 
0.52  793 
0.52  747 
0.52  701 
0.52  654 

9.98  174 
9.98  170 
9.98  166 
9.98  162 
9.98  159 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.45  547 
9.45  589 
9.45  632 
9.45  674 
9.45  716 

9.47  392 
9.47  438 
9.47  484 
9.47  530 
9.47  576 

0.52  608 
0.52  562 
0.52  516 
0.52  470 
0.52  424 

9.98  155 
9.98  151 
9.98  147 
9.98  144 
9.98  140 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.45  758 
9.45  801 
9.45  843 
9.45  885 
9.45  927 

9.47  622 
9.47  668 
9.47  714 
9.47  760 
9.47  806 

0.52  378 
0.52  332 
0.52  286 
0.52  240 
0.52  194 

9.98  136 
9.98  132 
9.98  129 
9.98  125 
9.98  121 

20 

19 
18 
17 
16 

45 

46 
47 
48 
49 

9.45  969 
9.46  Oil 
9.46  053 
9.46  095 
9.46  136 

9.47  852 
9.47  897 
9.47  943 
9.47  989 
9.48  035 

0.52  148 
0.52  103 
0.52  057 
0.52  Oil 
0.51  965 

9.98  117 
9.98  113 
9.98  110 
9.98  106 
9.98  102 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54. 

9.46  178 
9.46  220 
9.46  262 
9.46  303 
9.46  345 

9.48  080 
9.48  126 
9.48  171 
9.48  217 
9.48  262 

0.51  920 
0.51  874 
0.51  829 
0.51  783 
0.51  738 

9.98  098 
9.98  094 
9.98  090 
9.98  087 
9.98  083 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.46  386 
9.46  428 
9.46  469 
9.46  511 
9.46  552 

9.48  307 
9.48  353 
9.48  398 
9.48  443 
9.48  489 

0.51  693 
0.51  647 
0.51  602 
0.51  557 
0.51  511 

9.98  079 
9.98  075 
9.98  071 
9.98  067 
9.98  063 

5 
4 
3 
2 
1 

60 

9.46  594 

9.48  534 

0.51  466 

9.98  060 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

i 

[68] 


i 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

i 

2 
3 

4 

9.46  594 
9.46  635 
9.46  676 
9.46  717 
9.46  758 

9.48  534 
9.48  579 
9.48  624 
9.48  669 
9.48  714 

0.51  466 
0.51  421 
0.51  376 
0.51  331 
0.51  286 

9.98  060 
9.98  056 
9.98  052 
9.98  048 
9.98  044 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.46  800 
9.46  841 
9.46  882 
9.46  923 
9.46  964 

9.48  759 
9.48  804 
9.48  849 
9.48  894 
9.48  939 

0.51  241 
0'51  196 
0.51  151 
0.51  106 
0.51  061 

9.98  040 
9.98  036 
9.98  032 
9.98  029 
9.98  025 

55 
54 
53 
52 
51 

10 

11 
12 
13 

14 

9.47  005 
9.47  045 
9.47  086 
9.47  127 
9.47  168 

9.48  984 
9.49  029 
9.49  073 
9.49  118 
9.49  163 

0.51  016 
0.50  971 
0.50  927 
0.50  882 
0.50  837 

9.98  021 
9.98  017 
9.98  013 
9.98  009 
9.98  005 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.47  209 
9.47  249 
9.47  290 
9-47  330 
9.47  371 

9.49  207 
9.49  252 
9.49  296 
9.49  341 
9.49  385 

0.50  793 
0.50  748 
0.50  704 
0.50  659 
0.50  615 

9.98  001 
9.97  997 
9.97  993 
9.97  989 
9.97  986 

45 
44 
43 
42 
41 

20 

21 

22 
23 
24 

9.47  411 
9.47  452 
9.47  492 
9.47  533 
9.47  573 

9.49  430 
9.49  474 
9.49  519 
9.49  563 
9.49  607 

0.50  570 
0.50  526 
0.50  481 
0.50  437 
0.50  393 

9.97  982 
9.97  978 
9.97  974 
9.97  970 
9.97  966 

40 

39 
38 
37 
36 

72° 

17C 

25 
26 
27 
28 
29 

9.47  613 
9.47  654 
9.47  694 
9.47  734 
9.47  774 

9.49  652 
9.49  696 
9.49  740 
9.49  784 
9.49  828 

0.50  348 
0.50  304 
0.50  260 
0.50  216 
0.50  172 

9.97  962 
9.97  958 
9.97  954 
9.97  950 
9.97  946 

35 
34 
33 
32 
31 

•  t 

30 

31 
32 
33 
34 

9.47  814 
9.47  854 
9.47  894 
9.47  934 
9.47  974 

9.49  872 
9.49  916 
9.49  960 
9.50  004 
9.50  048 

0.50  128 
0.50  084 
0.50  040 
0.49  996 
0.49  952 

9.97  942 
9.97  938 
9.97  934 
9.97  930 
9.97  926 

30 

29 
28 

27 
26 

35 
36 
37 
38 
39 

9.48  014 
9.48  054 
9.48  094 
9.48  133 
9.48  173 

9.50  092 
9.50  136 
9.50  180 
9.50  223 
9.50  267 

0.49  908 
0.49  864 
0.49  820 
0.49  777 
0.49  733 

9.97  922 
9.97  918 
9.9?  914 
9.97  910 
9.97  906 

25 
24 
23 
2f2 
21 

40 

41 
42 
43 
44 

9.48  213 
9.48  252 
9.48  292 
9.48  332 
9.48  371 

9.50  311 
9.50  355 
9.50  398 
9.50  442 
9.50  485 

0.49  689 
0.49  645 
0.49  602 
0.49  558 
0.49  515 

9.97  902 
9.97  898 
9.97  894 
9.97  890 
9.97  886 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.48  411 
9.48  450 
9.48  490 
9.48  529 
9.48  568 

9.50  529 
9.50  572 
9.50  616 
9.50  659 
9.50  703 

0.49  471 
0.49  428 
0.49  384 
0.49  341 
0.49  297 

9.97  882 
9.97  878 
9.97  874 
9.97  870 
9.97  866 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.48  607 
9.48  647 
9.48  686 
9.48  725 
9.48  764 

9.50  746 
9.50  789 
9.50  833 
9.50  876 
9.50  919 

0.49  254 
0.49  211 
0.49  167 
0.49  124 
0.49  081 

9.97  861 
9.97  857 
9.97  853 
9.97  849 
9.97  845 

10 

9 
8 
7 
6 
5 
4 
3 
2 
1 

55 
56 
57 
58 
59 

9.48  803 
9.48  842 
9.48  881 
9.48  920 
9.48  959 

9.50  962 
9.51  005 
9.51  048 
9.51  092 
9.51  135 

0.49  038 
0.48  995 
0.48  952 
0.48  908 
0.48  865 

9.97  841 
9.97  837 
9.97  833 
9.97  829 
9.97  825 

60 

9.48  998 

9.51  178 

0.48  822 

997  821 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.Sin. 

p 

[59] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

71° 

0 

i 

2 
3 
4 

9.48  998 
9.49  037 
9.49  076 
9.49  115 
9.49  153 

9.51  178 
9.51  221 
9.51  264 
9.51  306 
9.51  349 

0.48  822 
0.48  779 
0.48  736 
0.48  694 
0.48  651 

9.97  821 
9.97  817 
9.97  812 
9.97  808 
9.97  804 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.49  192 
9.49  231 
9.49  269 
9.49  308 
9.49  347 

9.51  392 
9.51  435 
9.51  478 
9.51  520 
9.51  563 

0.48  608 
0.48  565 
0.48  522 
0.48  480 
0.48  437 

9.97  800 
9.97  796 
9.97  792 
9.97  788 
9.97  784 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.49  385 
9.49  424 
9.49  462 
9.49  500 
9.49  539 

9.51  606 
9.51  648 
9.51  691 
9.51  734 
9.51  776 

0.48  394 
0.48  352 
0.48  309 
0.48  266 
0.48  224 

9.97  779 
9.97  775 
9.97  771 
9.97  767 
9.97  763 

60 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.49  577 
9.49  615 
9.49  654 
9.49  692 
9.49  730 

9.51  819 
9.51  861 
9.51  903  - 
9.51  946 
9.51  988 

0.48  181 
0.48  139 
0.48  097 
0.48  054 
0.48  012 

9.97  759 
9.97  754 
9.97  750 
9.97  746 
9.97  742 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.49  768 
9.49  806 
9.49  844 
9.49  882 
9.49  920 

9.52  031 
9.52  073 
9.52  115 
9.52  157 
9.52  200 

0.47  969 
0.47  927 
0.47  885 
0.47  843 
0.47  800 

9.97  738 
9.97  734 
9.97  729 
9.97  725 
9.97  721 

40 

39 
38 
37 
36 

18° 

25 
26 
27 
28 
29 

9.49  958 
9.49  996 
9.50  034 
9.50  072 
9.50  110 

9.52  242 
9.52  284 
9.52  326 
9.52  368 
9.52  410 

0.47  758 
0.47  716 
0.47  674 
0.47  632 
0.47  590 

9.97  717 
9.97  713 
9.97  708 
9.97  704 
9.97  700 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.eO  148 
9.50  185 
9.50  223 
9.50  261 
9.50  298 

9.52  452 
9.52  494 
9.52  536 
9.52  578 
9.52  620 

0.47  548 
0.47  506 
0.47  464 
0.47  422 
0.47  380 

9.97  696 
9.97691 
9.97  687 
9.97  683 
9.97  679 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.50  336 
9.50  374 
9.50  111 
9.50  449 
9.50  486 

9.52  661 
9.52  703 
9.52  745 
9.52  787 
9.52  829 

0.47  339 
0.47  297 
0.47  255 
0.47  213 
0.47  171 

9.97  674 
9.97  670 
9.97  666 
9.97  662 
9.97  657 

25 
24 
23 
22 
21 

40 

41 

42 
43 
44 

9.50  523 
9.50  561 
9.50  598 
9.50  635 
9.50  673 

9.52  870 
9.52  912 
9.52  953 
9.52  995 
9.53  037 

0.47  130' 
0.47  088 
0.47  047 
0.47  005 
0.46  963 

9.97  653 
9.97  649 
9.97  645 
9.97  640 
9.97  636 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.50  710 
9.50  747 
9.50  784 
9.50  821 
9.50  858 

9.53  078 
9.53  120 
9.53  161 
9.53  202 
9.53  244 

0.46  922 
0.46  880 
0.46  839 
0.46  798 
0.46  756 

9.97  632 
9.97  628 
9.97  623 
9.97  619 
9.97  615 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.50  896 
9.50  933 
9.50  970 
9.51  007 
9.51  043 

9.53  285 
9.53  327 
9.53  368 
9.53  409 
9.53  450 

0.46  715 
0.46  673 
0.46  632 
0.46  591 
0.46  550 

9.97  610 
9.97  606 
9.97  602 
9.97  597 
9.97  593 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.51  080 
9.51  117 
9.51  154 
9.51  191 
9.51  227 

9.53  492 
9.53  533 
9.53  574 
9.53  615 
9.53  656 

0.46  508 
0.46  467 
0.46  426 
0.46  385 
0.46  344 

9.97  589 
9.97  584 
9.97  580 
9.97  576 
9.97  571 

5 
4 
3 
2 
1 

60 

9.51  264 

9.53  697 

0.46  303 

9.97  567 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[60] 


19° 

/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

1 
2 
3 
4 

9.51  264 
9.51  301 
9.51  338 
9.51  374 
9.51  411 

9.53  697 
9.53  738 
9.53  779 
9.53  820 
9.53  861 

0.46  303 
0.46  262 
0.46  221 
0.46  180 
0.46  139 

9.97  567 
9.97  563 
9.97  558 
9.97  554 
9.97  550 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.51  447 
9.51  484 
9.51  520 
9'51  557 
9.51  593 

9.53  902 
9.53  943 
9.53  984 
9.54  025 
9.54  065 

0.46  098 
0.46  057 
0.46  016- 
0.45  975 
0.45  935 

9.97  545 
9.97  541 
9.97  536 
9.97  532 
9.97  528 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.51  629 
9.51  666 
9.51  702 
9.51  738 
9.51  774 

9.54  106 
9.54  147 
9.54  187 
9.54  228 
9.54  269 

0.45  894 
0.45  853 
0.45  813 
0.45  772 
0.45  731 

9.97  523 
9.97  519 
9.97  515 
9.97  510 
9.97  506 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.51  811 
9.51  847 
9.51  883 
9.51  919 
9.51  955 

9.54  309 
9.54  350 
9.54  390 
9.54  431 
9.54  471 

0.45  691 
0.45  650 
0.45  610 
0.45  569 
0.45  529 

9.97  501 
9.97  497 
9.97  492 
9.97  488 
9.97  484 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.51  991 
9.52  027 
9.52  063 
9.52  099 
9.52  135 

9.54  512 
9.54  552 
9.54  593 
9.54  633 
9.54  673 

0.45  488 
0.45  448 
0.45  407 
0.45  367 
0.45  327 

9.97  479 
9.97  475 
9.97  470 
9.97  466 
9.97  461 

40 

39 
38 
37 
36 

70° 

25 

26 
27 
28 
29 

9.52  171 
9.52  207 
9.52  242 
9.52  278 
9.52  314 

9.54  714 
9.54  754 
9.54  794 
9.54  835 
9.54  875 

0.45  286 
0.45  246 
0.45  206 
0.45  165 
0.45  125 

9.97  457 
9.97  453 
9.97  448 
9.97  444 
9.97  439 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.52  350 
9.52  385 
9.52  421 
9.52  456 
9.52  492 

9.54  915 
9.54  955 
9.54  995 
9.55  035 
9.55  075 

0.45  085 
0.45  045 
0.45  005 
0.44  965 
0.44  925 

9.97  435 
9.97  430 
9.97  426 
9.97  421 
9.97  417 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.52  527 
9.52  563 
9.58  598 
9.52  634 
9.52  669 

9.55  115 
9.55  155 
9.55  195 
9.55  235 
9.55  275 

0.44  885 
0.44  845 
0.44  805 
0.44  765 
0.44  725 

9.97  412 
9.97  408 
9.97  403 
9.97  399 
9.97  394 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.52  705 
9.52  740 
9.52  775 
9.52  811 
9.52  846 

9.55  315 
9.55  355 
9.55  395 
9.55  434 
9.55  474 

0.44  685 
0.44  645 
0.44  605 
0.44  566 
0.44  526 

9.97  390 
9.97  385 
9.97  381 
9.97  376 
9.97  372 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.52  881 
9.52  916 
9.52  951 
9.52  986 
9.53  021 

9.55  514 
9.55  554 
9.55  593 
9.55  633 
9.55  673 

0.44  486 
0.44  446 
0.44  407 
0.44  367 
0.44  327 

9.97  367 
9.97  363 
9.97  358 
9.97  353 
9.97  349 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.53  056 
9.53  092 
9.53  126 
9.53  161 
9.53  196 

9.55  712 
9.55  752 
9.55  791 
9.55  831 
9.55  870 

0.44  288 
0.44  248 
0.44  209 
0.44  169 
0.44  130 

9.97  344 
9.97  340 
9.97  335 
9.97  331 
9.97  326 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.53  231 
9.53  266 
9.53  301 
9.53  336 
9.53  370 

9.55  910 
9.55  949 
9.55  989 
9.56  028 
9.56  067 

0.44  090 
0.44  051 
0.44  Oil 
0.43  972 
0.43  933 

9.97  322 
9.97  317 
9.97  312 
9.97  308 
9.97  303 

5 
4 
3 
2 
1 

60 

9.53  405 

9.56  107 

0.43  893 

9.97  299 

0 

' 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

; 

[61] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

69° 

0 

i 

2 
3 
4 

9.53  405 
9.53  440 
9.53  475 
9.53  509 
9.53  544 

9.56  107 
9.56  146 
9.56  185 
9.56  224 
9.56  264 

0.43  893 
0.43  854 
0.43  815 
0.43  776 
0.43  736 

9.97  299 
9.97  294 
9.97  289 
9.97  285 
9.97  280 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.53  578 
9.53  613 
9.53  647 
9.53  682 
9.53  716 

9.56  303 
9.56  342 
9.56  381 
9.56  420 
9.56  459 

0.43  697 
0.43  658 
0.43  619 
0.43  580 
0.43  541 

9.97  276 
9.97  271 
9.97  266 
9.97  262 
9.97  257 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.53  751 
9.53  785 
9.53  819 
9.53  854 
9.53  888 

9.56  498 
9.56  537 
9.56  576 
9.56  615 
9.56  654 

0.43  502 
0.43  463 
0.43  424 
0.43  385 
0.43  346 

9.97  252 
9.97  248 
9.97  243 
9.97  238 
9.97  234 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.53  922 
9.53  957 
9.53  991 
9.54  025 
9.54  059 

9.56  693 
9.56  732 
9.56  771 
9.56  810 
9.56  849 

0.43  307 
0.43  268 
0.43  229 
0.43  190 
0.43  151 

9.97  229 
9.97  224 
9.97  220 
9.97  215 
9.97  210 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.54  093 
9.54  127 
9.54  161 
9.54  195 
9.54  229 

9.56  887 
9.56  926 
9.56  965 
9.57  004 
9.57  042 

0.43  113 
0.43  074 
0.43  035 
0.42  996 
0.42  958 

9.97  206 
9.97  201 
9.97  196 
9.97  192 
9.97  187 

40 

39 
38 
37 
36 

20° 

25 
26 
27 
28 
29 

9.54  263 
9.54  297 
9.54  331 
9.54  365 
9.54  399 

9.57  081 
9.57  120 
9.57  158 
9.57  197 
9.57  235 

0.42  919 
0.42  880 
0.42  842 
0.42  803 
0.42  765 

9.97  182 
9.97  178 
9.97  173 
9.97  168 
9.97  163 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.54  433 
9.54  466 
9.54  500 
9.54  534 
9.54  567 

9.57  274 
9.57  312 
9.57  351 
9.57  389 
9.57  428 

0.42  726 
0.42  688 
0.42  649 
0.42  611 
0.42  572 

9.97  159 
9.97  154 
9.97  149 
9.97  145 
9.97  140 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.54  601 
9.54  635 
9.54  668 
9.54  702 
9.54  735 

9.57  466 
9.57  504 
9.57  543 
9.57  581 
9.57  619 

0.42  534 
0.42  496 
6.42  457 
0.42  419 
0.42  381 

9.97  135 
9.97  130 
9.97*126 
9.97  121 
9.97  116 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.54  769 
9.54  802 
9.54  836 
9.54  869 
9.54  903 

9.57  658 
9.57  696 
9.57  734 
9.57  772 
9.57  810 

0.42  342 
0.42  304 
0.42  266 
0.42  228 
0.42  190 

9.97  111 
9.97  107 
9.97  102 
9.97  097 
9.97  092 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.54  936 
9.54  969 
9.55  003 
9.55  036 
9.55  069 

9.57  849 
9.57  887 
9.57  925 
9.57  963 
9.58  001 

0.42  151 
0.42  113 
0.42  075 
0.42  037 
0.41  999 

9.97  087 
9.97  083 
9.97  078 
9.97  073 
9.97  068 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.55  102 
9.55  136 
9.55  169 
9.55  202 
9.55  235 

9.58  039 
9.58  077 
9.58  115 
9.58  153 
9.58  191 

0.41  961 
0.41  923 
0.41  885 
0.41  847 
0.41  809 

9.97  063 
9.97  059 
9.97  054 
9.97  049 
9.97  044 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.55  268 
9.55  301 
9.55  334 
9.55  367 
9.55  400 

9.58  229 
9.58  267 
9.58  304 
9.58  342 
9.58  380 

0.41  771 
0.41  733 
0.41  696 
0.41  658 
0.41  620 

9.97  039 
9.97  035 
9.97  030 
9.97  025 
9.97  020 

5 
4 
3 
2 
1 

60 

9.55  433 

9.58  418  - 

0.41  582 

9.97  015 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

f 

[62] 


9  jo 

t 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

1 
2 
3 
4 

9.55  433 
9.55  466 
9.55  499 
9.55  532 
9.55  564 

9.58  418 
9.58  455 
9.58  493 
9.58  531 
9.58  569 

0.41  582 
0.41  545 
0.41  507 
0.41  469 
0.41  431 

9.97  015 
9.97  010 
9.97  005 
9.97  001 
9.96  996 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.55  597 
9.55  630 
9.55  663 
9.55  695 
9.55  728 

9.58  606 
9.58  644 
9.58  681 
9.58  719 
9.58  757 

0.41  394 
0.41  356 
0.41  319 
0.41  281 
0.41  243 

9.96  991 
9.96  986 
9.96  981 
9.96  976 
9.96  971 

55 

54 
53 
52 
51 

68° 

10 

11 
12 
13 
14 

9.55  761 
9.55  793 
9.55  826 
9.55  858 
9.55  891 

9.58  794 
9.58  832 
9.58  869 
9.58  907 
9.58  944 

0.41  206 
0.41  168 
0.41  131 
0.41  093 
0.41  056 

9.96  966 
9.96  962 
9.96  957 
9.96  952 
9.96  947 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.55  923 
9.55  956 
9.55  988 
9.56  021 
9.56  053 

9.58  981 
9.59  019 
9.59  056 
9.59  094 
9.59  131 

0.41  019 
0.40  981 
0.40  944 
0.40  906 
0.40  869 

9.96  942 
9.96  937 
9.96  932 
9.96  927 
9.96  922 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.56  085 
9.56  118 
9.56  150 
9.56  182 
9.56  215 

9.59  168 
9.59  205 
9.59  243 
9.59  280 
9.59  317 

0.40  832 
0.40  795 
0.40  757 
0.40  720 
0.40  683 

9.96  917 
9.96  912 
9.96  907 
9.96  903 
9.96  898 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

9.56  247 
9.56  279 
9.56  311 
9.56  343 
9.56  375 

9.59  354 
9.59  391 
9.59  429 
9.59  466 
9.59  503 

0.40  646 
0.40  609 
0.40  571 
0.40  534 
0.40  497 

9.96  893 
9.96  888 
9.96  883 
9.96  878  • 
9.96  873 

35 

34 
33 
32 
31 

30 

31 

32 
33 
34 

9.56  408 
9.56  440 
9.56  472 
9.56  504 
9.56  536 

9.59  540 
9.59  577 
9.59  614 
9.59  651 
9.59  688 

0.40  460 
0.40  423 
0.40  386 
0.40  349 
0.40  312 

9.96  868 
9.96  863 
9.96  858 
9.96  853 
9.96  848 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.56  568 
9.56  599 
9.56  631 
9.56  663 
9.56  695 

9.59  725 
9.59  762 
9.59  799 
9.59  835 
9.59  872 

0.40  275 
0.40  238 
0.40  201 
0.40  165 
0.40  128 

9.96  843 
9.96  838 
9.96  833 
9.96  828 
9.96  823 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.56  727 
9.56  759 
9.56  790 
9.56  822 
9.56  854 

9.59  909 
9.59  946 
9.59  983 
9.60  019 
9.60  056 

0.40  091 
0.40  054 
0.40  017 
0.39  981 
0.39  944 

9.96  818 
9.96  813 
9.96  808 
9.96  803 
9.96  798 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.56  886 
9.56  917 
9.56  949 
9.56  980 
9.57  012 

9.60  093 
9.60  130 
9.60  166 
9.60  203 
9.60  240 

0.39  907 
0.39  870 
0.39  834 
0.39  797 
0.39  760 

9.96  793 
9.96  788 
9.96  783 
9.96  778 
9.96  772 

15 

14 
13 
12 
11 

50 

51 
52 
53 
54 

9.57  044 
9.57  075 
9.57  107 
9.57  138 
9.57  169 

9.60  276 
9'60  313 
9.60  349 
9.60  386 
9.60  422 

0.39  724 
0.39  687 
0.39  651 
0.39  614 
0.39  578 

9.96  767 
9.96  762 
9.96  757 
9.96  752 
9.96  747 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.57  201 
9.57  232 
9.57  264 
9.57  295 
9.57  326 

9.60  459 
9.GO  495 
9.60  532 
9.60  568 
9.60  605 

0.39  541 
0.39  505 
0.39  468 
0.39  432 
0.39  395 

9.96  742 
9.96  737 
9.96  732 
9.96  727 
9.96  722 

5 
4 
3 
2 
1 

60 

9.57  358 

9.«0  R41 

0.39  359 

9.96  717 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

; 

[63] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

1 
2 
3 
4 

9.57  358 
9.57  389 
9.57  420 
9.57  451 
9.57  482 

9.60  641 
9.60  677 
9.60  714 
9.60  750 
9.60  786 

0.39  359 
0.39  323 
0.39  286 
0.39  250 
0.39  214 

9.96  717 
9.96  711 
9.96  706 
9.96  701 
9.96  696 

60 

59 
58 
57 
56 

67° 

5 
6 
7 
8 
9 

9.57  514 
9.57  545 
9.57  576 
9.57  607 
9.57  638 

9.60  823 
9.60  859 
9.60  895 
9.60  931 
9.60  967 

0.39  177 
0.39  141 
0.39  105 
0.39  069 
0.39  033 

9.96  691 
9.96  686 
9.96  681 
9.96  676 
9.96  670 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.57  669 
9.57  700 
9.57  731 
9.57  762 
9.57  793 

9.61  004 
9.61  040 
9.61  076 
9.61  112 
9.61  148 

0.38  996 
0.38  960 
0.38  924 
0.38  888 
0.38  852 

9.96  665 
9.96  660 
9.96  655 
9.96  650 
9.96  645 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.57  824 
9.57  855 
9.57  885 
9.57  916 
9.57  947 

9.61  184 
9.61  220 
9.61  256 
9.61  292 
9.61  328 

0.38  816 
0.38  780 
0.38  744 
0.38  708 
0.38  672 

9.96  640 
9.96  634 
9.96  629 
9.96  624 
9.96  619 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.57  978 
9.58  008 
9.58  039 
9.58  070 
9.58  101 

9.61  364 
9.61  400 
9.61  436 
9.61  472 
9.61  508 

0.38  636 
0.38  600 
0.38  564 
0.38  528 
0.38  492 

9.96  614 
9.96  608 
9.96  603 
9.96  598 
9.96  593 

40 

39 
38 
37 
36 

99° 

25 
26 
27 
28 
29 

9.58  131 
9.58  162 
9.58  192 
*  9.58  223 
9.58  253 

9.61  544 
9.61  579 
9.61  615 
9.61  651 
9.61  687 

0.38  456 
0.38  421 
0.38  385 
0.38  349 
0.38  313 

9.96  588 
9.96  582 
9.96  577 
9.96  572 
9.96  567 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.58  284 
9.58  314 
9.58  345 
9.58  375 
9.58  406 

9.61  722 
9.61  758 
9.61  794 
9.61  830 
9.61  865 

0.38  278 
0.38  242 
0.38  206 
0.38  170 
0.38  135 

9.96  562 
9.96  556 
9.96  551 
9.96  546 
9.96  541 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.58  436 
9.58  467 
9.58  497 
9.58  527 
9.58  557 

9.61  901 
9.61  936 
9.61  972 
9.62  008 
9.62  043 

0.38  099 
0.38  064 
0.38  028 
0.37  992 
0.37  957 

9.96  535 
9.96  730 
9.96  525 
9.96  520 
9.96  514 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.58  588 
9.58  618 
9.58  648 
9.58  678 
9.58  709 

9.62  079 
9.62  114 
9.62  150 
9.62  185 
9.62  221 

0.37  921 
0.37  886 
0.37  850 
0.37  815 
0.37  779 

9.96  509 
9.96  504 
9.96  498 
9.96  493 
9.96  488 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.58  739 
9.58  769 
9.58  799 
9.58  829 
9.58  859 

9.62  256 
9.62  292 
9.62  327 
9.62  362 
9.62  398 

0.37  744 
0.37  708 
0.37  673 
0.37  638 
0.37  602 

9.96  483 
9.96  477 
9.96  472 
9.96  467 
9.96  461 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.58  889 
9.58  919 
9.58  949 
9.58  979 
9.59  009 

9.62  433 
9.62  468 
9.62  504 
9.62  539 
9.62  574 

0.37  567 
0.37  532  ' 
0.37  496 
0.37  461 
0.37  426 

9.96  456 
9.96  451 
9.96  445 
9.96  440 
9.96  435 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.59  039 
9.59  069 
9.59  098 
9.59  128 
9.59  158 

9.62  609 
9.62  645 
9.62  680 
9.62  715 
9.62  750 

0.37  391 
0.37  355 
0.37  320 
0.37  285 
0.37  250 

9.96  429 
9.96  424 
9.96  419 
9.96  413 
9.96  408 

5 
4 
3 
2 
1 

60 

9.59  188 

9.62  785 

0.37  215 

996403 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[64] 


r 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

66° 

0 

1 
2 
3 

4 

9.59  188 
9.59  218 
9.59  247 
9.59  277' 
9.59  307 

9.62  785 
9.62  820 
9.62  855 
9.62  890 
9.62  926 

0.37  215 
0.37  180 
0.37  145 
0.37  110 
0.37  074 

9.96  403 
9.96  397 
9.96  392 
9.96  387 
9.96  381 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.59  336 
9.59  366 
9.59  396 
9.59  425 
9.59  455 

9.62  961 
9.62  996 
9.63  031 
9.63  066 
9.63  101 

0.37  039 
0.37  004 
0.36  969 
0.36  934 
0.36  899 

9.96  376 
9.96  370 
9.96  365 
9.96  360 
9.96  354 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.59  484 
9.59  514 
9.59  543 
9.59  573 
9.59  602 

9.63  135 
9.63  170 
9.63  205 
9.63  240 
9.63  275 

0.36  865 
0.36  830 
0.36  795 
0.36  760 
0.36  725 

9.96  349 
9.96  343 
9.96  338 
9.96  333 
9.96  327 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.59  632 
9.59  661 
9.59  690 
9.59  720 
9.59  749 

9.63  310 
9.63  345 
9.63  379 
9.63  414 
9.63  449 

0.36  690 
0.36  655 
0.36  621 
0.36  586 
0.36  551 

9.96  322 
9.96  316 
9.96  311 
9.96  305 
9.96  300 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.59  778 
9.59  808 
9.59  837 
9.59  866 
9.59  895 

9.63  484 
9.63  519 
9.63  553 
9.63  588 
9.63  623 

0.36  516 
0.36  481 
0.36  447 
0.36  412 
0.36  377 

9.96  294 
9.96  289 
9.96  284 
9.96  278 
9.96  273 

40 

39 
38 
37 
36 

23° 

25 
26 
27 
28 
29 

9.59  924 
9.59  954 
9.59  983 
9.60  012 
9.60  041 

9.63  657 
9.63  692 
9.63  726 
9.63  761 
9.63  796 

0.36  343 
0.36  308 
0.36  274 
0.36  239 
0.36  204 

9.96  267 
9.96  262 
9.96  256 
9.96  251 
9.96  245 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.60  070 
9.60  099 
9.60  128 
9.60  157 
9.60  186 

9.63  830 
9.63  865 
9.63  899 
9.63  934 
9.63  968 

0.36  170 
0.36  135 
0.36  101 
0.36  066 
0.36  032 

9.96  240 
9.96  234 
9.96  229 
9.96  223 
9.96  218 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.60  215 
9.60  244 
9.60  273 
9.60  302 
9.60  331 

9.64  003 
9.64  037 
9.64  072 
9.64  106 
9.64  140 

0.35  997 
0.35  963 
0.35  928 
0.35  894 
0.35  860 

9.96  212 
9.96  207 
9.96  201 
9.96  196 
9.96  190 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.60  359 
9.60  388 
9.60  417 
9.60  446 
9.60  474 

9.64  175 
9.64  209 
9.64  243 
9.64  278 
9.64  312 

0.35  825 
0.35  791 
0.35  757 
0.35  722 
0.35  688 

9.96  185 
9.96  179 
9.96  174 
9.96  168 
9.96  162 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.60  503 
9.60  532 
9.60  561 
9.60  589 
9.60  618 

9.64  346 
9.64  381 
9.64  415 
9.64  449 
9.64  483 

0.35  654 
0.35  619 
0.35  585 
0.35  551 
0.35  517 

9.96  157 
9.96  151 
9.96  146 
9.96  140 
9.96  135 

15 

14 
13 
12 
11 

50 

51 
52 
53 
54 

9.60  646 
9.60  675 
9.60  704 
9.60  732 
9.60  761 

9.64  517 
9.64  552 
9.64  586 
9.64  620 
9.64  654 

0.35  483 
0.35  448 
0.35  414 
0.35  380 
0.35  346 

9.96  129 
9.96  123 
9.96  118 
9.96  112 
9.96  107 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.60  789 
9.60  818 
9.60  846 
9.60  875 
9.60  903 

9.64  688 
9.64  722 
9.64  756 
9.64  790 
9.64  824 

0.35  312 
0.35  278 
0.35  244 
0.35  210 
0.35  176 

9.96  101 
9.96  095 
9.96  090 
9.96  084 
9.96  079 

5 
4 
3 
2 
1 

60 

9.60  931 

9.64  858 

0.35  142 

9.96  073 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

; 

[65] 


1 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

65° 

• 

0 

1 

2 
3 
4 

9.60  931 
9.60  960 
9.60  988 
9.61  016 
9.61  045 

9.64  858 
9.64  892 
9.64  926 
9.64  960 
9.64  994 

0.35  142 
0.35  108 
0.35  074 
0.35  040 
0.35  006 

9.96  073 
9.96  067 
•  9.96  062 
9.96  056 
9.96  050 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.61  073 
9.61  101 
9.61129 
9.61  158 
9.61  186 

9.65  028 
9.65  062 
9.65  096 
9.65  130 
9.65  164 

0.34  972 
0.34  938 
0.34  904 
0.34  870 
0.34  836 

9.96  045 
9.96  039 
9.96  034 
9.96  028 
9.96  022 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.61  214 
9.61  242 
9.61  270 
9.61  298 
9.61  326 

9.65  197 
9.65  231 
9.65  265 
9.65  299 
9.65  333 

0.34  803 
0.34  769 
0.34  735 
0.34  701 
0.34  667 

9.96  017 
9.96  Oil 
9.96  005 
9.96  000 
9.95  994 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.61  354 
9.61  382 
9.61  411 
9.61  438 
9.61  466 

9.65  366 
9.65  400 
9.65  434 
9.65  467 
9.65  501 

0.34  634 
0.34  600 
0.34  566 
0.34  533 
0.34  499 

9.95  988 
9.95  982 
9.95  977 
9.95  971 
9.95  965 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.61  494 
9.61  522 
9.61  550 
9.61  578 
9.61  606 

9.65  535 
9.65  568 
9.65  602 
9.65  636 
9.65  669 

0.34  465 
0.34  432 
0.34  398 
0.34  364 
0.34  331 

9.95  960 
9.95  954 
9.95  948 
9.95  942 
9.95  937 

40 

39 
38 
37 
36 

24° 

25 
26 
27 
28 
29 

9.61  634 
9.61  662 
9.61  689 
9.61  717 
9.61  745 

9.65  703 
9.65  736 
9.65  770 
9.65  803 
9.65  837 

0.34  297 
0.34  264 
0.34  230 
0.34  197 
0.34  163 

9.95  931 
9.95  925 
9.95  920 
9.95  914 
9.95  908 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.61  773 
9.61  800 
9.61  828 
9.61  856 
9.61  883 

9.65  870 
9.65  904 
9.65  937 
9.65  971 
9.66  004 

0.34  130 
0.34  096 
0.34  063 
0.34  029 
0.33  996 

9.95  902 
9.95  897 
9.95  891 
9.95  885 
9.95  879 

30 

29 

28 
27 
26 

35 
36 
37 
38 
39 

9.61  911 
9.61  939 
9.61  966 
9.61  994 
9.62  021 

9.66  038 
9.66  071 
9.66  104 
9.66  138 
9.66  171 

0.33  962 
0.33  929 
0.33  896 
0.33  862 
0.33  829 

9.95  873 
9.95  868 
9.95  862 
9.95  856 
9.95  850 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.62  049 
9.62  076 
9.62  104 
9.62  131 
9.62  159 

9.66  204 
9.66  238 
9.66  271 
9.66  304 
9.66  337 

0.33  796 
0.33  762 
0.33  729 
0.33  696 
0.33  663 

9.95  844 
9.95  839 
9.95  833 
9.95  827 
9.95  821 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.62  186 
9.62  214 
9.62  241 
9.62  268 
9.62  296 

9.66  371 
9.66  404 
9.66  437 
9.66  470 
9.66  503 

0.33  629 
0.33  596 
0.33  563 
0.33  530 
0.33  497 

9.95  815 
9.95  810 
9.95  804 
9.95  798 
9.95  792 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.62  323 
9.62  350 
9.62  377 
9.62  405 
9.62  432 

9.66  537 
9.66  570 
9.66  603 
9.66  636 
9.66  669 

0.33  463 
0.33  430 
0.33  397 
0.33  364 
0.33  331 

9.95  786 
9.95  780 
9.95  775 
9.95  769 
9.95  763 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.62  459 
9.62  486 
9.62  513 
9.62  541 
9.62  568 

9.66  702 
9.66  735 
9.66  768 
9.66  801 
9.66  834 

0.33  298 
0.33  265 
0.33  232 
0.33  199 
0.33  166 

9.95  757 
9.95  751 
9.95  745 
9.95  739 
9.95  733 

5 
4 
3 
2 
1 

60 

9.62  595 

9.66  867 

0.33  133 

9.95  728 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[66] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

64° 

0 

1 

2 
3 
4 

9.62  595 
9.62  622 
9.62  649 
9.62  676 
9.62  703 

9.66  867 
9.66  900 
9.66  933 
9.66  966 
9.66  999 

0.33  133 
0.33  100 
0.33  067 
0.33  034 
0.33  001 

9.95  728 
9.95  722 
9.95  716 
9.95  710 
9.95  704 

60 

59 
53 
57 
56 

5 
6 
7 
8 
9 

9.62  730 
9.62  757 
9.62  784 
9.62  811 
9.62  838 

9.67  032 
9.67  065 
9.67  098 
9.67  131 
9.67  163 

0.32  968 
0.32  935 
0.32  902 
0.32  869 
0.32  837 

9.95  698 
9.95  692 
9.95  686 
9.95  680 
9.95  674 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.62  865 
9.62  892 
9.62  918 
9.62  945 
9.62  972 

9.67  196 
9.67  229 
9.67  262 
9.67  295 
9.67  327 

0.32  804 
0.32  771 
0.32  738 
0.32  705 
0.32  673 

9.95  668 
9.95  663 
9.95  657 
9.95  651 
9  95  645 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.62  999 
9.63  026 
9.63  052 
9.63  079 
9.63  106 

9.67  360 
9.67  393 
9.67  426 
9.67  458 
9.67  491 

0.32  640 
0.32  607 
0.32  574 
0.32  542 
0.32  509 

9.95  639 
9.95  633 
9.95  627 
9.95  621 
9.95  615 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.63  133 
9.63  159 
9.63  186 
9.63  213 
9.63  239 

9.67  524 
9.67  556 
9.67  589 
9.67  622 
9.67  654 

0.32  476 
0.32  444 
0.32  411 
0.32  378 
0.32  346 

9.95  609 
9.95  603 
9.95  597 
9.95  591 
9.95  585 

40 

39 

38 
37 
36 

25° 

25 
26 
27 
28 
29 

9.63  266 
9.63  292 
9.63  319 
9.63  345 
9.63  372 

9.67  687 
9.67  719 
9.67  752 
9.67  785 
9.67  817 

0.32  313 
0.32  281 
0.32  248 
0.32  215 
0.32  183 

9.85  579 
9.95  573 
9.95  567 
9.95  561 
9.95  555 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.63  398 
9.63  425 
9.63  451 
9.63  478 
9.63  504 

9.67  850 
9.67  882 
9.67  915 
9.67  947 
9.67  980 

0.32  150 
0.32  118 
0.32  085 
0.32  053 
0.32  020 

9.95  549 
9.95  543 
9.95  537 
9.95  531 
9.95  525 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.63  531 
9.63  557 
9.63  583 
9.63  610 
9.63  636 

9.68  012 
9.68  044 
9.68  077 
9.68  109 
9.68  142 

0.31  988 
0.31  956 
0.31  923 
0.31  891 
0.31  858 

9.95  519 
9.95  513 
9.95  507 
9.95  500 
9.95  494 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.63  662 
9.63  689 
9.63  715 
9.63  741 
9.63  767 

9.68  174 
9.68  206 
9.68  239 
9.68  271 
9.68  303 

0.31  826 
0.31  794 
0.31  761 
0.31  729 
0.31  697 

9.95  488 
9.95  482 
9.95  476 
9.95  470 
9.95  464 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.63  794 
9.63  820 
9.63  846 
9.63  872 
9.63  898 

9.68  336 
9.68  368 
9.68  400 
9.68  432 
9.68  465 

0.31  664 
0.31  632 
0.31  600 
0.31  568 
0.31  535 

9.95  458 
9.95  452 
9.95  446 
9.95  440 
9.95  434 

15 

14 
13 
12 
11 

60 

51 
52 
53 
54 

9.63  924 
9.63  950 
9.63  976 
9.64  002 
9.64  028 

9.68  497 
9.68  529 
9.68  561 
9.68  593 
9.68  626 

0.31  503 
0.31  471 
0.31  439 
0.31  407 
0.31  374 

9.95  427 
9.95  421 
9.95  415 
9.95  409 
9.95  403 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.64  054 
9.64  080 
9.64  106 
9.64  132 
9.64  158 

9.68  658 
9.68  690 
9.68  722 
9.68  754 
9.68  786 

0.31  342 
0.31  310 
0.31  278 
0.31  246 
0.31  214 

9.95  397 
9.95  391 
9.95  384 
9.95  378 
9.95  372 

5 
4 
3 
2 
1 

60 

9.64  184 

9.68  818 

0.31  182 

9.95  366 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang.- 

L.  Sin. 

; 

[67] 


2fi° 

/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

i 

2 
3 
4 

9.64  184 
9.64  210 
9.64  236 
9.64  262 
9.64  288 

9.68  818 
9.68  850 
9.68  882 
9.68  914 
9.68  946 

0.31  182 
0.31  150 
0.31  118 
0.31  086 
0.31  054 

9.95  366 
9.95  360 
9.95  354 
9.95  348 
9.95  341 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.64  313 
9.64  339 
9.64  365 
9.64  391 
9.64  417 

9.68  978 
9.69  010 
9.69  042 
9.69  074 
9.69  106 

0.31  022 
0.30  990 
0.30  958 
0.30  926 
0.30  894 

-  9.95  335 
9.95  329 
9.95  323 
9.95  317 
9.95  310 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.64  442 
9.64  468 
9.64  494 
9.64  519 
9.64  545 

9.69  138 
9.69  170 
9.69  202 
9.69  234 
9.69  266 

0.30  862 
0.30  830 
0.30  798 
0.30  766 
0.30  734 

9.95  304 
9.95  298 
9.95  292 
9.95  286 
9.95  279 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.64  571 
9.64  596 
9.64  622 
9.64  647 
9.64  673 

9.69  298 
9.69  329 
9.69  361 
9.69  393 
9.69  425 

0.30  702 
0.30  671 
0.30  639 
0.30  607 
0.30  575 

9.95  273 
9.95  267 
9.95  261 
9.95  254 
9.95  248 

45 
44 
43 
42 
41 

20 

21 
22 

23 
24 

9.64  698 
9.64  724 
9.64  749 
9.64  775 
9.64  800 

9.69  457 
9.69  488 
9.69  520 
9.69  552 
9.69  584 

0.30  543 
0.30  512 
0.30  480 
0.30  448 
0.30  416 

9.95  242 
9.95  236 
9.95  229 
9.95  223 
9.95  217 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

9.64  826 
9.64  851 
9.64  877 
9.64  902 
9.64  927 

9.69  615 
9.69  647 
9.69  679 
9.69  710 
9.69  742 

0.30  385 
0.30  353 
0.30  321 
0.30  290 
0.30  258 

9.95  211 
9.95  204 
9.95  198 
9.95  192 
9.95  185 

35 
34 
33 
32 
31 

63° 

«\j 

30 

31 
32 
33 
34 

9.64  953 
9.64  978 
9.65  003 
9.65  029 
9.65  054 

9.69  774 
9.69  805 
9.69  837 
9.69  868 
9.69  900 

0.30  226 
0.30  195 
0.30  163 
0.30  132 
0.30  100 

9.95  179 
9.95  173 
9.95  167 
9.95  160 
9.95  154 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.65  079 
9.65  104 
9.65  130 
9.65  155 
9.65  180 

9.69  932 
9.69  963 
9.69  995 
9.70  026 
9.70  058 

0.30  068 
0.30  037 
0.30  005 
0.29  974 
0.29  942 

9.95  148 
9.95  141 
9.95  135 
9.95  129 
9.95  122 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.65  205 
9.65  230 
9.65  255 
9.65  281 
9.65  306 

9.70  089 
9.70  121 
9.70  152 
9.70  184 
9.70  215 

0.29  911 
0.29  879 
0.29  848 
0.29  816 
0.29  785 

9.95  116 
9.95  110 
9.95  103 
9.95  097 
9.95  090 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.65  331 
9.65  356 
9.65  381 
9.65  406 
9.65  431 

9.70  247 
9.70  278 
9.70  309 
9.70  341 
9.70  372 

0.29  753 
0.29  722 
0.29  691 
0.29  659 
0.29  628 

9.95  084 
9.95  078 
9.95  071 
9.95  065 
9.95  059 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.65  456 
9.65  481 
9.65  506 
9.65  531 
9.65  556 

9.70  404 
9.70  435 
9.70  466 
9.70  498 
9.70  529 

0.29  596 
0.29  565 
0.29  534 
0.29  502 
0.29  471 

9.95  052 
9.95  046 
9.95  039 
9.95  033 
9.95  027 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.65  580 
9.65  605 
9.65  630 
9.65  655 
9.65  680 

9.70  560 
9.70  592 
9.70  623 
9.70  654 
9.70  685 

0.29  440 
0.29  408 
0.29  377 
0.29  346 
0.29  315 

9.95  020 
9.95  014 
9.95  007 
9.95  001 
9.94  995 

5 
4 
3 
2 
1 

60 

9.65  705 

9.70  717 

0.29  283 

9.94  988 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[68] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

1 
2 
3 
4 

9.65  705 
9.65  729 
9.65  754 
9.65  779 
9.65  804 

9.70  717 
9.70  748 
9.70  779 
9.70  810 
9.70  841 

0.29  283 
0.29  252 
0.29  221 
0.29  190 
0.29  159 

9.94  988 
9.94  982 
9.94  975 
9.94  969 
9.94  962 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.65  828 
9.65  853 
9.65  878 
9.65  902 
9.65  927 

9.70  873 
9.70  904 
9.70  935 
9.70  966 
9.70  997 

0.29  127 
0.29  096 
0.29  065 
0.29  034 
0.29  003 

9.94  956 
9.94  949 
9.94  943 
9.94  936 
9.94  930 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.65  952 
9.65  976 
9.66  001 
9.66  025 
9.66  050 

9.71  028 
9.71  059 
9.71  090 
9.71  121 
9.71  153 

0.28  972 
0.28  941 
0.28  910 
0.28  879 
0.28  847 

9.94  923 
9.94  917 
9.94  911 
9.94  904 
9.94  898 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.66  075 
9.66  099 
9.66  124 
9.66  148 
9.66  173 

9.71  184 
9.71  215 
9.71  246 
9.71  277 
9.71  308 

0.28  816 
0.28  785 
0.28  754 
0.28  723 
0.28  692 

9.94  891 
9.94  885 
9.94  878 
9.94  871 
9.94  865 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.66  197 
9.66  221 
9.66  246 
9.66  270 
9.66  295 

9.71  339 
9.71  370 
9.71  401 
9.71  431 
9.71  462 

0.28  661 
0.28  630 
0.28  599 
0.28  569 
0.28  538 

9.94  858 
9.94  852 
9.94  845 
9.94  839 
9.94  832 

40 

39 
38 
37 
36 

27° 

25 
26 
27 
28 
29 

9.66  319 
9.66  343 
9.66  368 
9.66  392 
9.66  416 

9.71  493 
9.71  524 
9.71  555 
9.71  586 
9.71  617 

0.28  507 
0.28  476 
0.28  445 
0.28  414 
0.28  383 

9.94  826 
9.94  819 
9.94  813 
9.94  806 
9.94  799 

35 
34 
33 
32 
31 

62° 

30 

31 
32 
33 
34 

9.66  441 
9.66  465 
9.66  489 
9.66  513 
9.66  537 

9.71  648 
9.71  679 
9.71  709 
9.71  740 
9.71  771 

0.28  352 
0.28  321 
0.28  291 
0.28  260 
0.28  229 

9.94  793 
9.94  786 
9.94  780 
9.94  773 
9.94  767 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.66  562 
9.66  586 
9.66  610 
9.66  634 
9.66  658 

9.71  802 
9.71  833 
9.71  863 
9.71  894 
9.71  925 

0.28  198 
0.28  167 
0.28  137 
0.28  106 
0.28  075 

9.94  760 
9.94  753 
9.94  747 
9.94  740 
9.94  734 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.66  682 
9.66  706 
9.66  731 
9.66  755 
9.66  779 

9.71  955 
9.71  986 
9.72  017 
9.72  048 
9.72  078 

0.28  045 
0.28  014 
0.27  983 
0.27  952 
0.27  922 

9.94  727 
9.94  720 
9.94  714 
9.94  707 
9.94  700 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.66  803 
9.66  827 
9.66  851 
9.66  875 
9.66  899 

9.72  109 
9.72  140 
9.72  170 
9.72  201 
9.72  231 

0.27  891 
0.27  860 
0.27  830 
0.27  799 
0.27  769   ' 

9.94  694 
9.94  687 
9.94  680 
9.94  674 
9.94  667 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.66  922 
9.66  946 
9.66  970 
9.66  994 
9.67  018 

9.72  262 
9.72  293 
9.72  323 
9.72  354 
9.72  384 

.  0.27  738 
0.27  707 
0.27  677 
0.27  646 
0.27  616 

9.94  660 
9.94  654 
9.94  647 
9.94  640 
9.94  634 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.67  042 
9.67  066 
9.67  090 
9.67  113 
9.67  137 

9.72  415 
9.72  445 
9.72  476 
9.72  506 
9.72  537 

0.27  585 
0.27  555 
0.27  524 
0.27  494 
0.27  463 

9.94-627 
9.94  620 
9.94  614 
9.94  607 
9.94  600 

5 
4 
3 
2 
1 

60 

9.67  161 

9.72  567 

0.27  433 

9.94  593 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[69] 


i 

I.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

i 

2 
3 

4 

9.67  161 
9.67  185 
9.67  208 
9.67  232 
9.67  256 

9.72  567 
9.72  598 
9.72  628 
9.72  659 
9.72  689 

0.27  433 
0.27  402 
0.27  372 
0.27  341 
0.27311 

9.94  593 
9.94  587 
9.94  580 
9.94  573 
9.94  567 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.67  280 
9.67  303 
9.67  327 
9.67  350 
9.67  374 

9.72  720 
9.72  750 
9.72  780 
9.72  811 
9.72  841 

0.27  280 
0.27  250 
0.27  220 
0.27  189 
0.27  159 

9.94  560 
9.94  553 
9.94  546 
9.94  540 
9.94  533 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.67  398 
9.67  421 
9.67  445 
9.67  468 
9.67  492 

9.72  872 
9.72  902 
9.72  932 
9.72  963 
9  72  993 

0.27  128 
0.27  098 
0.27  068 
0.27  037 
0.27  007 

9.94  526 
9.94  519 
9.94  513 
9.94  506 
9.94  499 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.67  515 
9.67  539 
9.67  562 
9.67  586 
9.67  609 

9.73  023 
9.73  054 
9.73  084 
9.73  114 
9.73  144 

0.26  977 
0.26  946 
0.26  916 
0.26  886 
0.26  856 

9.94  492 
9.94  485 
9.94  479 
9.94  472 
9.94  465 

45 
44 
43 
42 
41 

61° 

20 

21 
22 
23 

24 

9.67  633 
9.67  656 
9.67  680 
9.67  703 
9.67  726 

9.73  175 
9.73  205 
9.73  235 
9.73  265 
9.73  295 

0.26  825 
0.26  795 
0.26  765 
0.26  735 
0.26  705 

9.94  458 
9.94  451 
9.94  445 
9.94  438 
9.94  431 

40 

39 
38 
37 
36 

28° 

25 
26 
27 
28 
29 

9.67  750 
9.67  773 
9.67  796 
9.67  820 
9.67  843 

9.73  326 
9.73  356 
9.73  386 
9.73  416 
9.73  446 

0.26  674 
0.26  644 
0.26  614 
0.26  584 
0.26  554 

9.94  424 
9.94  417 
9.94  410 
9.94  404 
9.94  397 

35 
34 
33 
32 
31 

30 

31 
32 

33 
34 

9.67  866 
9.67  890 
9.67  913 
9.67  936 
9.67  959 

9.73  476 
9.73  507 
9.73  537 
9.73  567 
9.73  597 

0.26  524 
0.26  493 
0.26  463 
0.26  433 
0.26  403 

9.94  390 
9.94  383 
9.94  376 
9.94  369 
9.94  362 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.67  982 
9.68  006 
9.68  029 
9.68  052 
9.68  075 

9.73  627 
9.73  657 
9.73  687 
9.73  717 
9.73  747 

0.26  373 
0.26  343 
0.26  313 
0.26  283 
0.26  253 

9.94  355 
9.94  349 
9.94  342 
9.94  335 
9.94  328 

25 
24 
23 
22 

21 

40 

41 
42 
43 
44 

9.68  098 
9.68  121 
9.68  144 
9.68  167 
9.68  190 

9.73  777 
9.73  807 
9.73  837 
9.73  867 
9.73  897 

0.26  223 
0.26  193 
0.26  163 
0.26  133 
0.26  103 

9.94  321 
9.94  314 
9.94  307 
9.94  300 
9.94  293 

20 

19 
18 

17 
16 

45 
46 
47 
48 
49 

9.68  213 
9.68  237 
9.68  260 
9.68  283 
9.68  305 

9.73  927 
9.73  957 
9.73  987 
9.74  017 
9.74  047 

0.26  073 
0.26  043 
0.26  013 
0.25  983 
0.25  953 

9.94  286 
9.94  279 
9.94  273 
9.94  266 
9.94  259 

15 
14  . 
13 
12 
11 

50 

51 
52 
53 
54 

9.68  328 
9.68  351 
9.68  374 
9.68  397 
9.68  420 

9.74  077 
9.74  107 
9.74  137 
9.74  166 
9.74  196 

0.25  923 
0.25  893 
0.25  863 
0.25  834 
0.25  804 

9.94  252 
9.94  245 
9.94  238 
9.94  231 
9.94  224 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.68  443 
9.68  466 
9.68  489 
9.68  512 
9.68  534 

9.74  226 
9.74  256 
9.74  286 
9.74  316 
9.74  345 

0.25  774 
0.25  744 
0.25  714 
0.25  684 
0.25  655 

9.94  217 
9.94  210 
9.94  203 
9.94  196 
9.94  189 

5 
4 
3 
2 
1 

60 

9.68  557 

9.74  375 

0.25  625 

9.94  182 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[70] 


29° 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

r 

60° 

0 

i 

2 
3 
4 

9.68  557 
9.68  580 
9.68  603 
9.68  625 
9.68  648 

9.74  375 
9.74  405 
9.74  435 
9.74  465 
9.74  494 

0.25  625 
0.25  595 
0.25  565 
0.25  535 
0.25  506 

9.94  182 
9.94  175 
9.94  168 
9.94  161 
9.94  154 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.68  671 
9.68  694 
9.68  716 
9.68  739 
9.68  762 

9.74  524 
9.74  554 
9.74  583 
9.74  613 
9.74  643 

0.25  476 
0.25  446 
0.25  417 
0.25  387 
0.25  357 

9.94  147 
9.94  140 
9.94  133 
9.94  126 
9.94  119 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.68  784 
9.68  807 
9.68  829 
9.68  852 
9.68  875 

9.74  673 
9.74  702 
9.74  732 
9.74  762 
9.74  791 

0.25  327 
0.25  298 
0.25  268 
0.25  238 
0.25  209 

9.94  112 
9.94  105 
9.94  098 
9.94  090 
9.94  083 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.68  897 
9.68  920 
9.68  942 
9.68  965 
9.68  987 

9.74  821 
9.74  851 
9.74  880 
9.74  910 
9.74  939 

0.25  179 
0.25  149 
0.25  120 
0.25  090 
0.25  061 

9.94  076 
9.94  069 
9.94  062 
9.94  055 
9.94  048 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.69  010 
9.69  032 
9.69  055 
9.69  077 
9.69  100 

9.74  969 
9.74  998 
9.75  028 
9.75  058 
9.75  087 

0.25  031 
0.25  002 
0.24  972 
0.24  942 
0.24  913 

9.94  041 
9.94  034 
9.94  027 
9.94  020 
9.94  012 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

9.69  122 
9.69  144 
9.69  167 
9.69  189 
9.69  212 

9.75  117 
9.75  146 
9.75  176 
9.75  205 
9  75  235 

0.24  883 
0.24  854 
0.24  824 
0.24  795 
0.24  765 

9.94  005 
9.93  998 
9.93  991 
9.93  984 
9.93  977 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.69  234 
9.69  256 
9.69  279 
9.69  301 
9.69  323 

9.75  264 
9.75  294 
9.75  323 
9.75  353 
9.75  382 

0.24  736 
0.24  706 
0.24  677 
0.24  647 
0.24  618 

9.93  970  . 
9.93  963 
9.93  955 
9.93  948 
9.93  941 

30 

29 
28 
27 
26 

35 
36  . 
37 
38 
39 

9.69  345 
9.69  368 
9.69  390 
9.69  412 
9.69  434 

9.75  411 
9.75  441 
9.75  470 
9.75  500 
9.75  529 

0.24  589 
0.24  559 
0.24  530 
0.24  500 
0.24  471 

9.93  934 
9.93  927 
9.93  920 
9.93  912 
9.93  905 

25 
24 
23 
22 
21 

f 

42 

:43 
-44 

9.69  356 
9.69  479 
9.69  501 
9.69  523 
9.69  545 

9.75  558 
9.75  588 
9.75  617 
9.75  647 
9.75  676 

0.24  442 
0.24  412 
0.24  383 
0.24  353 
0.24  324 

9.93  898 
9.93  891 
9.93  884 
9.93  876 
9.93  869 

20 

19 
18 
17 
16 

45 
46 
"47 

48 
49 

9.69  567 
9.69  589 
9.69  611 
9.69  633 
9.69  655 

9.75  705 
9.75  735 
9.75  764 
9.75  793 
9.75  822 

0.24  295 
0.24  265 
0.24  236 
0.24  207 
0.24  178 

9.93  862 
9.93  855 
9.93  847 
9.93  840 
9.93  833 

15 
14 
13 
12 
11 

50 

51 
52 
53 

54 

9.69  677 
9.69  699 
9.69  721 
9.69  743 
9.69  765 

9.75  852 
9.75  881 
9.75  910 
9.75  939 
9.75  969 

0.24  148 
0.24  119 
0.24  090 
0.24  061 
0.24  031 

9.93  826 
9.93  819 
9.93  811 
9.93  804 
9.93  797 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.69  787 
9.69  809 
9.69  831 
9.69  853 
9.69  875 

9.75  998 
9.76  027 
9.76  056 
9.76  086 
9.76  115 

0.24  002 
0.23  973 
0.23  944 
0.23  914 
0.23  885 

9.93  789 
9.93  782 
9.93  775 
9.93  768 
9.93  760 

5 
4 
3 
2 
1 

60 

9.69  897 

9.76  144 

0.23  856 

9.93  753 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

r 

[71] 


' 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

1 

2 
3 

4 

9.69  897 
9.69  919 
9.69  941 
9.69  963 
9.69  984 

9.76  144 
9.76  173 
9.76  202 
9.76  231 
9.76  261 

0.23  856 
0.23  827 
0.23  798 
0.23  769 
0.23  739 

9.93  753 
9.93  746 
9.93  738 
9.93  731 
9.93  724 

60  1 

59 
58 

1 

5 
6 
7 
8 
9 

9.70  006 
9.70  028 
9.70  050 
9.70  072 
9.70  093 

9.76  290 
9.76  319 
9.76  348 
9.76  377 
9.76  406 

0.23  710 
0.23  681 
0.23  652 
0.23  623 
0.23  594 

9.93  717 
9.93  709 
9.93  702 
9.93  695 
9.93  687 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.70  115 
9.70  137 
9.70  159 
9.70  180 
9.70  202 

9.76  435 
9.76  464 
9.76  493 
9.76  522 
9.76  551 

0.23  565 
0.23  536 
0.23  507 
0.23  478 
0.23  449 

9.93  680 
9.93  673 
9.93  665 
9.93  658 
9.93  650 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.70  224 
9.70  245 
9.70  267 
9.70  288 
9.70  310 

9.76  580 
9.76  609 
9.76  639 
9.76  668 
9.76  697 

0.23  420 
0.23  391 
0.23  361 
0.23  332 
0.23  303 

9.93  643 
9.93  636 
9.93  628 
9.93  621 
9.93  614 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.70  332 
9.70  353 
9.70  375 
9.70  396 
9.70  418 

9.76  725 
9.76  754 
9.76  783 
9.76  812 
9.76  841 

0.23  275 
0.23  246 
0.23  217 
0.23  188 
0.23  159 

9.93  606 
9.93  599 
9.93  591 
9.93  584 
•  9.93  577 

40 

39 
38 
37 
36 

OA° 

25 
26 
27 
28 
29 

9.70  439 
9.70  461 
9.70  482 
9.70  504 
9.70  525 

9.76  870 
9.76  899 
9.76  928 
9.76  957 
9.76  986 

0.23  130 
0.23  101 
0.23  072 
0.23  043 
0.23  014 

9.93  569 
9.93  562 
9.93  554 
9.93  547 
9.93  539 

35 
34 
33 
32 
31 

Ovf 

30 

31 
32 
33 
34 

9.70  547 
9.70  568 
9.70  590 
9.70  611 
9.70  633 

9.77  015 
9.77  044 
9.77  073 
9.77  101 
9.77  130 

0.22  985 
0  22  956 
0.22  927 
0.22  899 
0.22  870 

9.93  532 
9.93  525 
9.93  517 
9.93  510 
9.93  502 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.70  654 
9.70  675 
9.70  697 
9.70  718 
9.70  739 

9.77  159 
9.77  188 
9.77  217 
9.77  246 
9.77  274 

0.22  841 
0.22  812 
0.22  783 
0.22  754 
0.22  726 

9.93  495 
9.93  487 
9.93  480 
9.93  472 
9.93  465 

25 
.  24 
23 
22 
21 

40 

41 
42 
43 
44 

9.70  761 
9.70  782 
9.70  803 
9.70  824 
9.70  846 

9.77  303 
9.77  332 
9.77  361 
9.77  390 
9.77  418 

0.22  697 
0.22  668 
'  0.22  639 
0.22  610 
0.22  582 

9.93  457 
9.93  450 
9.93  442 
9.93  435 
9.93  427 

20^ 

inj 

45 
46 
47 
48 
49 

9.70  867 
9.70  888 
9.70  909 
9.70  931 
9.70  952 

9.77  447 
9.77  476 
9.77  505 
9.77  533 
9.77  562 

0.22  553 
0.22  524 
0.22  495 
0.22  467 
0.22  438 

9.93  420 
9.93  412 
9.93  405 
9.93  397 
9.93  390 

14 
13 
12 
11 

50 

51 

52 
53 
54 

9.70  973 
9.70  994 
9.71  015 
9.71  036 
9.71  058 

9.77  591 
9.77  619 
9.77  648 
9.77  677 
9.77  706 

0.22  409 
0.22  381 
0.22  352 
0.22  323 
0.22  294 

9.93  382 
9.93  375 
9.93  367 
9.93  360 
9.93  352 

10» 

8 
7 
6 

55 
56 
57 
58 
59 

9.71  079 
9.71  100 
9.71  121 
9.71  142 
9  71  163 

9.77  734 
9.77  763 
9.77  791 
9.77  820 
9.77  849 

0.22  266 
0.22  237 
0.22  209 
0.22  180 
0.22  151 

9.93  344 
9.93  337 
9.93  329 
9.93  322 
9.93  314 

5 
4 
3 
2 
1 

60 

9.71-184 

9.77.  877 

0.22  123 

9.93  307 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

r 

[72] 


L.  Sin. 

9.71  184 
9.71  205 
9.71  226 
9.71  247 
).71  268 


9.71  289 
9.71  310 
9.71  331 
9.71  352 
9.71  373 


9.71  393 
9.71  414 
9.71  435 
9.71  456 
9.71  477 


9.72  421 
L.  Cos. 


L.  Tang. 

9.77  877 
9.77  906 
9.77  935 
9.77  963 
9.77  992 


9.78  020 
9.78  049 
9.78  077 
9.78  106 
9.78  135 


9.78  163 
9.78  192 
9.78  220 
9.78  249 
9.78  277 


9.78  306 
9.78  334 
9.78  363 
9.78  391 
9.78  419 


9.78  448 
9.78  476 
9.78  505 
9.78  533 
9.78  562 


9.78  590 
9.78  618 
9.78  647 
9.78  675 
9.78  704 


.71  498 
9.71  519 
9.71  539 
9.71  560 
9.71  581 


9.71  602 
9.71  622 
9.71  643 
9.71  664 
9.71  685 


9.71  705 
9.71  726 
9.71  747 
9.71  767 
9.71  788 


9.71  809 
.71  829 
9.71  850 
9.71  870 
9.71  891 


9.71  911 
9.71  932 
9.71  952 
9.71  973 
9.71  994 


9.72  014 
9.72  034 
9.72  055 
9.72  075 
9.72  096 


9.79  015 
9.79  043 
9.79  072 
9.79  100 
9.79  128 


9.72  116 
9.72  137 
9.72  157 
9.72  177 
9.72  198 


9.79  156 
9.79  185 
9.79  213 
9.79  241 
9.79  269 


9.72  218 
9.72  238 
9.72  259 
9.72  279 
9.72  299 


9.72  320 
9.72  340 
9.72  360 
9.72  381 
9.72  401 


9.78  732 
9.78  760 
9.78  789 
9.78  817- 
9.78  845 


).78  874 
9.78  902 
9.78  930 
9.78  959 
9.78  987 


9.79  297 
9.79  326 
9.79  354 
9.79  382 
9.79  410 


9.79  438 
9.79  466 
9.79  495 
9.79  523 
9.79  551 


9.79  579 


L.  Cotg. 


L.  Cotg. 

0.22  123 
0.22  094 
0.22  065 
0.22  037 
0.22  008 


0.21  980 
0.21  951 
0.21  923 
0.21  894 
0.21  865 


0.21  837 
0.21  808 
0.21  780 
0.21  751 
0.21  723 


0.21  694 
0.21  666 
0.21  637 
0.21  609 
0.21  581 


0.21  552 
0.21  524 
0.21  495 
0.21  467 
0.21  438 


0.21  410 
0.21  382 
0.21  353 
0.21  325 
0.21  296 


0.21  268 
0.21  240 
0.21  211 
0.21  183 
0.21  155 


0.21  126 
0.21  098 
0.21  070 
0.21  041 
0.21  013 


0.20  985 
0.20  957 
0.20  928 
0.20  900 
0.20  872 


0.20  844 
0.20  815 
0.20  787 
0.20  759 
0.20  731 


0.20  703 
0.20  674 
0.20  646 
0.20  618 
0.20  590 


0.20  562 
0.20  534 
0.20  505 
0.20  477 
0.20  449 


0.20  421 


L.  Tang. 


[73] 


L.  Cos. 

9.93  307 
9.93  299 
9.93  291 
9.93  284 
9.93  276 


9.93  269 
9.93  261 
9.93  253 
9.93  246 
9.93  238 


9.93  230 
9.93  223 
9.93  215 
9.93  207 
9.93  200 


9.93  192 
9.93  184 
9.93  177 
9.93  169 
9.93  161 


9.93  154 
9.93  146 
9.93  138 
9.93  131 
9.93  123 


9.93  115 
9.93  108 
9.93  100 
9.93  092 
9.93  084 


9.93  077 
9.93  069 
9.93061 

9.93  053 
9.93  046 


9.93  038 
9.93  030 
9.93  022 
9.93  014 
9.93  007 


9.92  999 
9.92  991 
9.92  983 
9.92  976 
9.92  968 


9.92  960 
9.92  952 
9.92  944 
9.92  936 
9.92  929 


9.92  921 
9.92  913 
9.92  905 
9.92  897 
9.92  889 


9.92  881 
9.92  874 
9.92  866 
9.92  858 
9.92  850 


9.92  842 
L.  Sin. 


60 

59 
58 

57 
56 


55 
54 
53 
52 
51 


50 

49 
48 
47 
46 


45 
44 
43 
42 
41 


40 

39 
38 
37 
36 


35 
34 
33 
32 
31 


30 

29 
28 
27 
26 


25 
24 
23 
22 
21 


20 

19 
18 
17 
16 


15 
14 
13 
12 
11 


10 

9 
8 
7 
6 


58° 


' 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

57° 

0 

i 

2 
3 
4 

9.72  421 
9.72  441 
9.72  461 
9.72  482 
9.72  502 

9.79  579 
9.79  607 
9.79  635 
9.79  663 
9.79  691 

0.20  421 
0.20  393 
0.20  365 
0.20  337 
0.20  309 

9.92  842 
9.92  834 
9.92  826 
9.92  818 
9.92  810 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.72  522 
9.72  542 
9.72  562 
9.72  582 
9.72  602 

9.79  719 
9.79  747 
9.79  776 
9.79  804 
9.79  832 

0.20  281 
0.20  253 
0.20  224 
0.20  196 
0.20  168 

9.92  803 
9.92  795 
9.92  787 
9.92  779 
9.92  771 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.72  622 
9.72  643 
9.72  663 
9.72  683 
9.72  703 

9.79  860 
9.79  888 
9.79  916 
9.79  944 
9.79  972 

0.20  140 
0.20  112 
0.20  084 
0.20  056 
0.20  028 

9.92  763 
9.92  755 
9.92  747 
9.92  739 
9.92  731 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.72  723 
9.72  743 
9.72  763 
9.72  783 
9.72  803 

9.80  000 
9.80  028 
9.80  056 
9.80  084 
9.80  112 

0.20  000 
0.19  972 
0.19  944 
0.19  916  ' 
0.19  888 

9.92  723 
9.92  715 
9.92  707 
9.92  699 
9.92  691 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.72  823 
9.72  843 
9.72  863 
9.72  883 
9.72  902 

9.80  140 
9.80  168 
9.80  195 
9.80  223 
9.80  251 

0.19  860 
0.19  832 
0.19  805 
0.19  777 
0.19  749 

9.92  683 
9.92  675 
9.92  667 
9.92  659 
9.92  651 

40 

39 
38 
37 
36 

32° 

25 
26 
27 
28 
29 

9.72  922 
9.72  942 
9.72  962 
9.72  982 
9.73  002 

9.80  279 
9.80  307 
9.80  335 
9.80  363 
9.80  391 

0.19  721 
0.19  693 
0.19  665 
0.19  637 
0.19  609 

9.92  643 
9.92  635 
9.92  627 
9.92  619 
9.92  611 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.73  022 
9.73  041 
9.73  061 
9.73  081 
9.73  101 

9.80  419 
9.80  447 
9.80  474 
9.80  502 
9.80  530 

0.19  581 
0.19  553 
0.19  526 
0.19  498 
0.19  470 

9.92  603 
9.92  595 
9.92  587-- 
9.92  579 
9.92  571 

30 

29 
28 
27 
26 

CO  CO  CO  CO  CO 
CO  OO  -<  CO  Ol 

V 

9.73  121 
9.73  140 
9.73  160 
9.73  180 
9.73  200 

9.80  558 
9.80  586 
9.80  614 
9.80  642 
9.80  669 

0.19  442 
0.19  414 
0.19  386 
0.19  358 
0.19  331 

9.92  563 
9.92  555 
9.92  546 
9.92  538 
9.92  530 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.73  219 
9.73  239 
9.73  259 
9.73  278 
9.73  298 

9.80  697 
9.80  725 
9.80  753 
9.80  781 
9.80  808 

0.19  303 
0.19  275 
0.19  247 
0.19  219 
0.19  192 

9.92  522 
9.92  514 
9.92  506 
9.92  498 
9.92  490 

20 

19, 
18 
17 

45 
46 
47 
48 
49 

9.73  318 
9.73  337 
9.73  357 
9.73  377 
9.73  396 

9.80  836 
9.80  864 
9.80  892 
9.80  919 
9.80  947 

0.19  164 
0.19  136 
0.19  108 
0.19  081 
0.19  053 

9.92  482 
9.92  473 
9.92  465 
9.92  457 
9.92  449 

15 
14 
13 
12 
11 

50 

51 
52 

53 
54 

9.73  416 
9.73  435 
9.73  455 
9.73  474 
9.73  494 

9.80  975 
9.81  003 
9.81  030 
9.81  058 
9.81  086 

0.19  025 
0.18  997 
0.18  970 
0.18  942 
0.18  914 

9.92  441 
9.92  433 
9.92  425 
9.92  416 
9.92  408 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.73  513 
9.73  533 
9.73  552 
9.73  572 
9.73  591 

9.81  113 
9.81  141 
9.81  169 
9.81  196 
9.81  224 

0.18  887 
0.18  859 
0.18  831" 
0.18  804 
0.18  776 

•9.92  400 
9.92  392 
9.92  384 
9.92  376 
9.92  367 

5 
4 
3 
2 
1 

60 

9.73  611 

9.81  252 

0.18  748 

9.92  359 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

t 

i 

L.  Siii. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

i 

2 
3 

4 

9.73  611 
9.73  630 
9.73  650 
9.73  669 
9.73  689 

9.81  252 
9.81  279 
9.81  307 
9.81  335 
9.81  362 

0.18  748 
0.18  721 
0.18  693 
0.18  665 
0.18  638 

9.92  359 
9.92  351 
9.92  343 
9.92  335 
9.92  326 

2 

58 

57 
56 

5 
6 
7 
8 
9 

9.73  708 
9.73  727 
9.73  747 
9.73  766 
9.73  785 

9.81  390 
9.81  418 
9.81  445 
9.81  473 
9.81  500 

0.18  610 
0.18  582 
0.18  555 
0.18  527 
0.18  500 

9.92  318 
9.92  310 
9.92  302 
9.92  293 
9.92  285 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.73  805 
9.73  824 
9.73  843 
9.73  863 
9.73  882 

9.81  528 
9.81  556 
9.81  583 
9.81  611 
9.81  638 

0.18  472 
0.18  444 
0.18  417 
0.18  389 
0.18  362 

9.92  277 
9.92  269 
9.92  260 
9.92  252 
9.92  244 

50  l^tol 

46* 

15 
16 
17 
18 
19 

9.73  901 
9.73  921 
9.73  940 
9.73  959 
9.73  978 

9.81  666 
9.81  693 
9.81  721 
9.81  748 
9.81  776 

0.18  334 
0.18  307 
0.18  279 
0.18  252 
0.18  224 

9.92  235 
9.92  227 
9.92  219 
9.92  211 
9.92  202 

45     A 

44 
43 
42 
41  '| 

20 

21 
22 
23 
24 

9.73  997 
9.74  017 
9.74  036 
9.74  055 
9.74  074 

9.81  803 
9.81  831 
9.81  858 
9.81  886 
9.81  913 

0.18  197 
0.18  169 
0.18  142 
0.18  114 
0.18  087 

9.92  194 
9.92  186 
9.92  177 
9.92  169 
9.92  161 

40 

39 
38 
37 
36  | 

33° 

25 
26 
27 
28 
29 

9.74  093 
9.74  113 
9.74  132 
9.74  151 
9.74  170 

9.81  941 
9.81  968 
9.81  996 
9.82  023 
9.82  051 

0.18  059 
0.18  032 
0.18  004 
0.17  977 
0.17  949 

9.92  152 
9.92  144 
9.92  136 
9.92  127 
9.92  119 

35 
34 
33 
32 

31  56° 

30 

31 
32 
33 
34 

9.74  189 
9.74  208 
9.74  227 
9.74  246 
9.74  265 

9.82  078 
9.82  106 
9.82  133 
9.82  161 
9.82  188 

0.17  922 
0.17  894 
0.17  867 
0.17  839 
0.17  812 

9.92  111 
9.92  102 
9.92  094 
9.92  086 
9.92  077 

30  Y 

29 
28 
27 
26  | 

35 
36 
37 
38 
39 

9.74  284 
9.74  303 
9.74  322 
9.74  341 
9.74  360 

9.82  215 
9.82  243 
9.82  270 
9.82  298 
9.82  325 

0'.17  785 
0.17  757 
0.17  730 
0.17  702 
0.17  675 

9.92  069 
9.92  060 
9.92  052 
9.92  044 
9.92  035 

25 
24 
23 
22 
21 

40 

41- 
42 
43 
44 

9.74  379 
9.74  398 
9.74  417 
9.74  436 
9.74  455 

9.82  352 
9.82  380 
9.82  407 
9.82  435 
9.82  462 

0.17  648 
0.17  620 
0.17  593 
0.17  565 
0.17  538 

9.92  027 
9.92  018 
9.92  010 
9.92  002 
9.91  993 

20 

19 
18 
17 
16  | 

45 
46 
47, 
48 
49 

9.74  474 
9.74  493 
9.74  512 
9.74  531 
9.74  549 

9.82  489 
9.82  517 
9.82  544 
9.82  571 
9.82  599 

0.17  511 
0.17  483 
0.17  456 
0.17  429 
0.17401 

9.91  985 
9.91  976 
9.91  968 
9.91  959 
9.91  951 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.74  568 
9.74  587 
9.74  606 
9.74  625 
9.74  644 

9.82  626 
9.82  653 
9.82  681 
9.82  708 
9.82  735 

0.17  374 
0.17  347 
0.17  319 
0.17  292 
0.17  265 

9.91  942 
9.91  934 
9.91  925 
9.91  917 
9.91  908 

10 

8 

I 

55 
56 
57 
58 
59 

9.74  662 
9.74  681 
9.74  700 
9.74  719 
9.74  737 

9.82  762 
9.82  790 
9.82  817 
9.82  844 
9.82  871 

0.17  238 
0.17  210 
0.17  183 
0.17  156 
0.17  129 

9.91  900 
9.91  891 
9.91  883 
9.91  874 
9.91  866 

35 
] 

60 

9.74  756 

9.82  899 

0.17  101 

9.91  857 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

; 

[75] 


31° 

/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

55° 

0 

i 

2 
3 
4 

9.74  756 
9.74  775 
9.74  794 
9.74  812 
9.74  831 

9.82  899 
9.82  926 
9.82  953 
9.82  980 
9.83  008 

0.17  101 
0.17  074 
0.17  047 
0.17  020 
0.16  992 

9.91  857 
9.91  849 
9.91  840 
9.91  832 
9.91  823 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.74  850 
9.74  868 
9.74  887 
9.74  906 
9.74  924  • 

9.83  035 
9.83  062 
9.83  089 
9.83  117 
9.83  144 

0.16  965 
0.16  938 
0.16  911 
0.16  883 
0.16  856 

9.91  815 
9.91  806 
9.91  798 
9.91  789 
9.91  781 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.74  943 
9.74  961 
9.74  980 
9.74  999 
9.75  017 

9.83  171 
9.83  198 
9.83  225 
9.83  252 
9.83  280 

0.16  829 
0.16  802 
0.16  775 
0.16  748 
0.16  720 

9.91  772 
9.91  763 
9.91  755 
9.91  746 
9.91  738 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.75  036 
9.75  054 
9.75  073 
9.75  091 
9.75  110 

9.83  307 
9.83  334 
9.83  361 
9.83  388 
9.83  415 

0.16  693 
0.16  666 
0.16  639 
0.16  612 
0.16  585 

9.91  729 
9.91  720 
9.91  712 
9.91  703 
9.91  695 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.75  128 
9.75  147 
9.75  165 
9.75  184 
9.75  202 

9.83  442 
9.83  470 
9.83  497 
9.83  524 
9.83  551 

0.16  558 
0.16  530 
0.16  503 
0.16  476 
0.16  449 

9.91  686 
9.91  677 
9.91  669 
9.91  660 
9.91  651 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

9.75  221 
9.75  239 
9.75  258 
9.75  276 
9.75  294 

9.83  578 
9.83  605 
9.83  632 
9.83  659 
9.83  686 

0.16  422 
0.16  395 
0.16  368 
0.16  341 
0.16  314 

9.91  643 
9.91  634 
9.91  625 
9.91  617 
9.91  608 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.75  313 
9.75  331 
9.75  350 
9.75  368 
9.75  386 

9.83  713 
9.83  740 
9.83  768 
9.83  795 
9.83  822 

0.16  287 
0.16  260 
0.16  232 
0.16  205 
0.16  178 

9.91  599 
9.91  591 
9.91  582 
9.91  573 
9.91  565 

30 

29 
28 
27 
26 

• 

35 
36 
37 
38 
39 

9.75  405 
9.75  423 
9.75  441 
9.75  459 
9.75  478 

9.83  849 
9.83  876 
9.83  903 
9.83  930 
9.83  957 

0.16  151 
0.16  124 
0.16  097 
0.16  070 
0.16  043 

9.91  556 
9.91  547 
9.91  538 
9.91  530 
9.91  521 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.75  496 
9.75  514 
9.75  533 
9.75  551 
9.75  569 

9.83  984 
9.84  Oil 
9.84  038 
9.84  065 
9.84  092 

0.16  016 
0.15  989 
0.15  962 
0.15  935 
0.15  908 

9.91  512 
9.91  504 
9.91  495 
9.91  486 
9.91  477 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.75  587 
9.75  605 
9.75  624 
9.75  642 
9.75  660 

9.84  119 
9.84  146  ' 
9.84  173 
9.84  200 
9.84  227 

0.15  881 
0.15  854 
0.15  827 
0.15  800 
0.15  773 

9.91  469 
9.91  460 
9.91  451 
9.91  442 
9.91  433 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.75  678 
9.75  696 
9.75  714 
9.75  733 
9.75  751 

9.84  254 
9.84  280 
9.84  307 
9.84  334 
9.84  361 

0.15  746 
0.15  720 
0.15  693 
0.15  666 
0.15  639 

9.91  425 
9.91  416 
9.91  407 
9.91  398 
9.91  389 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.75  769 
9.75  787 
9.75  805 
9.75  823 
9.75  841 

9.84  388 
9.84  415 
9.84  442 
9.84  469 
9.84  496 

0.15  612 
0.15  585 
0.15  558 
0.15  531 
0.15  504 

9.91  381 
9.91  372 
9.91  363 
9.91  354 
9.91  345 

5 
4 
3 
2 
1 

60 

9.75  859 

9.84  523 

0.15  477 

9.91  336 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

t 

[76] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

1 

2 
3 
4 

9.75  859 
9.75  877 
9.75  895 
9.75  913 
9.75  931 

9.84  523 
9.84  550 
9.84  576 
9.84  603 
9.84  630 

0.15  477 
0.15  450 
0.15  424  ' 
0.15  397 
0.15  370 

9.91  336 
9.91  328 
9.91  319 
9.91  310 
9.91  301 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.75  949 
9.75  967 
9.75  985 
9.76  003 
9.76  021 

9.84  657 
9.84  684 
9.84  711 
9.84  738 
9.84  764 

0.15  343 
0.15  316 
0.15  289 
0.15  262 
0.15  236 

9.91  292 
9.91  283 
9.91  274 
9.91  266 
9.91  257 

55 
54 
53 
52 
51 

54° 

10 

11 
12 
13 
14 

9.76  039 
9.76  057 
9.76  075 
9.76  093 
9.76  111 

9.84  791 
9.84  818 
9.84  845 
9.84  872 
9.84  899 

0.15  209 
0.15  182 
0.15  155 
0.15  128 
0.15  101 

9.91  248 
9.91  239 
9.91  230 
9.91  221 
9.91  212 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.76  129 
9.76  146 
9.76  164 
9.76  182 
9.76  200 

9.84  925 
9.84  952 
9.84  979 
9.85  006 
9.85  033 

0.15  075 
0.15  048 
0.15  021 
0.14  994 
0.14  967 

9.91  203 
9.91  194 
9.91  185 
9.91  176 
9.91  167 

45 
44 
43 
42 
41 

35° 

20 

21 
22 
23 
24 

9.76  218 
9.76  236 
9.76  253 
9.76  271 
9.76  289 

9.85  059 
9.85  086 
9.85  113 
9.85  140 
9.85  166 

0.14  941 
0.14  914 
0.14  887 
0.14  860 
0.14  834 

9.91  158 
9.91  149 
9.91  141 
9.91  132 
9.91  123 

40 

39 
38 
37 
36 

25 

26 
27 
28 
29 

9.76  307 
9.76  324 
9.76  342 
9.76  360 
9.76  378 

9.85  193 
9.85  220 
9.85  247 
9.85  273 
9.85  300 

0.14  807 
0.14  780 
0.14  753 
0.14  727 
0.14  700 

9.91  114 
9.91  105 
9.91  096 
9.91  087 
9.91  078 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.76  395 
9.76  413 
9.76  431 
9.76  448 
9.76  466 

9.85  327 
9.85  354 
9.85  380 
9.85  407 
9.85  434 

0.14  673 
0.14  646 
0.14  620 
0.14  593 
0.14  566 

9.91  069 
9.91  060 
9.91  051 
9.91  042 
9.91  033 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.76  484 
9.76  501 
9.76  519 
9.76  537 
9.76  554 

9.85  460 
9.85  487 
9.85  514 
9.85  540 
9.85  567 

0.14  540 
0.14  513 
0.14  486 
0.14  460 
0.14  433 

9.91  023 
9.91  014 
9.91  005 
9.90  996 
9.90  987 

25 
24 
23 
22 
21 

40 

41 

42 
43 
44 

9.76  572 
9.76  590 
9.76  607 
9.76  625 
9.76  642 

9.85  594 
9.85  620 
9.85  647 
9.85  674 
9.85  700 

0.14  406 
0.14  380 
0.14  353 
0.14  326 
0.14  300 

9.90  978 
9.90  969 
9.90  960 
9.90  951 
9.90  942 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.76  660 
9.76  677 
9.76  695 
9.76  712 
9.76  730 

9.85  727 
9.85  754 
9.85  780 
9.85  807 
9.85  834 

0.14  273 
0.14  246 
0.14  220 
0.14  193 
0.14  166 

9.90  933 
9.90  924 
9.90  915 
9.90  906 
9.90  896 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.76  747 
9.76  765 
9.76  782 
9.76  800 
9.76  817 

9.85  860 
9.85  887 
9.85  913 
9.85  940 
9.85  967 

0.14  140 
0.14  113 
0.14  087 
0.14  060 
0.14  033 

9.90  887 
9.90  878 
9.90  869 
9.90  860 
9.90  851 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.76  835 
9.76  852 
9.76  870 
9.76  887 
9.76  904 

9.85  993 
9.86  020 
9.86  046 
9.86  073 
9.86  100 

0.14  007 
0.13  980 
0.13  954 
0.13  927 
0.13  900 

9.90  842 
9.90  832 
9.90  823 
9.90  814 
9.90  805 

5 
4 
3 
2 
1 

60 

9.76  922 

9.86  126 

0.13  874 

9.90  796 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[77] 


1 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

53° 

0 
1 

2 
3 
4 

9.76  922 
9.76  939 
9.76  957 
9.76  974 
9.76  991 

9.86  126 
9.86  153 
9.86  179 
9.86  206 
9.86  232 

0.13  874 
0.13  847 
0.13  821 
0.13  794 
0.13  768 

9.90  796 
9.90  787 
9.90  777 
9.90  768 
9.90  759 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.77  009 
9.77  026 
9.77  043 
9.77  061 
9.77  078 

9.86  259 
9.86285 
9.86  312 
9.86  338 
9.86  365 

0.13  741 
0.13715 
0.13  688 
0.13  662 
0.13  635 

9.90  750 
9.90  741 
9.90  731 
9.90  722 
9.90  713 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.77  095 
9.77  112 
9.77  130 
9.77  147 
9.77  164 

9.86  392 
9.86  418 
9.86  445 
9.86  471 
9.86  498 

0.13  608 
0.13  582 
0.13  555 
0.13  529 
0.13  502 

9.90  704 
9.90  694 
9.90  685 
9.90  676 
9.90  667 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.77  181 
9.77  199 
9.77  216 
9.77  233 
9.77  250 

9.86  524 
9.86  551 
9.86  577 
9.86  603 
9.86  630 

0.13  476 
0.13  449 
0.13  423 
0.13  397 
0.13  370 

9.90  657 
9.90  648 
9.90  639 
9.90  630 
9.90  620 

45 
44 
43 
42 
41 

20 

21 
22 
23 

24* 

9.77  268 
9.77  285 
9.77  302 
9.77  319 
9.77  336 

9.86  656 
9.86  683 
9.86  709 
9.86  736 
9.86  762 

0.13  344 
0.13  317 
0.13  291 
0.13  264 
0.13  238 

9.90  611 
9.90  602 
9.90  592 
9.90  583 
9.90  574 

40 

39 
38 
37 
36 

36° 

25 
26 
27 
28 
29 

9.77  353 
9.77  370 
9.77  387 
9.77  405 
9.77  422 

9.86  789 
9.86  815 
9.86  842 
9.86  868 
9.86  894 

0.13  211 
0.13  185 
0.13  158 
0.13  132 
0.13  106 

9.90  565 
9.90  555 
9.90  546 
9.90  537 
9.90  527 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.77  439 
9.77  456- 
9.77  473 
9.77  490 
9.77  507 

9.86  921 
9.86  947 
9.86  974 
9.87  000 
9.87  027 

0.13  079 
0.13  053 
0.13  026 
0.13  000 
0.12  973 

9.90  518 
9.90  509 
9.90  499 
9.90  490 
9.90  480 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.77  524 
9.77  541 
9.77  558 
9.77  575 
9.77  592 

9.87  053 
9.87  079 
9.87  106 
9.87  132 
9.87  158 

0.12  947 
0.12  921 
0.12  894 
0.12  868 
0.12  842 

9.90  471 
9.90  462 
9.90  452 
9.90  443 
9.90  434 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.77  609 
9.77  626 
9.77  643 
9.77  660 
9.77  677 

9.87  185 
9.87  211 
9.87  238 
9.87  264 
9.87  290 

0.12  815 
0.12  789 
0.12  762 
0.12  736 
0.12  710 

9.90  424 
9.90  415 
9.90  405 
9.90  396 
9.90  386 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.77  694 
9.77  711 
9.77  728 
9.77  744 
9.77  761 

9.87  317 
9.87  343 
9.87  369 
9.87  396 
9.87  422 

0.12  683 
0.12  657 
0.12  631 
0.12  604 
0.12  578 

9.90  377 
9.90  368 
9.90  358 
9.90  349 
9.90  339 

15 
14 
13 
12 
11 

50 

51 

52 
53 
54 

9.77  778 
9.77  795 
9.77  812 
9.77  829 
9.77  846 

9.87  448 
9.87  475 
9.87  501 
9.87  527 
9.87  554 

0.12  552 
0.12  525 
0.12  499 
0.12  473 
0.12  446 

9.90  330 
9.90  320 
9.90  311 
9.90  301 
9.90  292 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.77  862 
9.77  879 
9.77  896 
9.77  913 
9.77  930 

9.87  580 
9.87  606 
9.87  633 
9.87  659 
9.87  685 

0.12  420 
0.12  394 
0.12  367 
0.12  341 
0.12  315 

9.90  282 
9.90  273 
9.90  263 
9.90  254 
9.90  244 

5 
4 
3 
2 
1 

60 

977  946 

9.87  711 

0.12  289 

9.90  235 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

r 

[78] 


37° 

/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

52° 

0 

1 
2 
3 
4 

9.77  946 
9.77  963 
9.77  980 
9.77  997 
9.78  013 

9.87  711 
9.87  738 
9.87  764 
9.87  790 
9.87  817 

0.12  289 
0.12  262 
0.12  236 
0.12  210 
0.12  183 

9.90  235 
9.90  225 
9.90  216 
9.90  206 
9.90  197 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.78  030 
9.78  047 
9.78  063 
9.78  080 
9.78  097 

9.87  843 
9.87  869 
9.87  895 
9.87  922 
9.87  948 

0.12  157 
0.12  131 
0.12  105 
0.12  078 
0.12  052 

9.90  187 
9.90  178 
9.90  168 
9.90  159 
9.90  149 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.78  113 
9.78  130 
9.78  147 
9.78  163 
9.78  180 

9.87  974 
9.88  000 
9.88  027 
9.88  053 
9.88  079 

0.12  026 
0.12  000 
0.11  973 
0.11  947 
0.11  921 

9.90  139 
9.90  130 
9.90  120 
9.90  111 
9.90  101 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.78  197 
9.78  213 
9.78  230 
9.78  246 
9.78  263 

9.88  105 
9.88  131 
9.88  158 
9.88  184 
9.88  210 

0.11  895 
0.11  869 
0.11  842 
0.11  816 
0.11  790 

9.90  091 
9.90  082 
9.90  072 
9.90  063 
9.90  053 

45 
44 
43 
42 
41 

20 

21 

22 
23 
24 

9.78  280 
9.78  296 
9.78  313 
9.78  329 
9.78  346 

9.88  236 
9.88  262 
9.88  289 
9.88  315 
9.88  341 

0.11  764 
0.11  738 
0.11  711 
0.11  685 
0.11  659 

9.90  043 
9.90  034 
9.90  024 
9.90  014 
9.90  005 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

9.78  362 
9.78  379 
9.78  395 
9.78  412 
9.78  428 

9.88  367 
9.88  393 
9.88  420 
9.88  446 
9.88  472 

0.11  633 
0.11  607 
0.11  580 
0.11  554 
0.11  528 

9.89  995 
9.89  985 
9.89  976 
9.89  966 
9.89  956 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.78  445 
9.78  461 
9.78  478 
9.78  494 
9.78  510 

9.88  498 
9.88  524 
9.88  550 
9.88  577 
9.88  603 

0.11  502 
0.11  476 
0.11  450 
0.11  423 
0.11  397 

9.89  947 
9.89  937 
9.89  927 
9.89  918 
9.89  908 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.78  527 
9.78  543 
9.78  560 
9.78  576 
9.78  592 

9.88  629 
9.88  655 
9.88  681 
9.88  707 
9.88  733 

0.11  371 
0.11  345 
0.11  319 
0.11-293 
0.11  267 

9.89  898 
9.89  888 
9.89  879 
9.89  869 
9.89  859 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.78  609 
9.78  625 
9.78  642 
9.78  658 
9.78  674 

9.88  759 
9.88  786 
9.88  812 
9.88  838 
9.88  864 

0.11  241 
0.11  214 
0.11  188 
0.11  162  ' 
0.11  136 

9.89  849 
9.89  840 
9.89  830 
9.89  820 
9.89  810 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.78  691 
9.78  707 
9.78  723 
9.78  739 
9.78  756 

9.88  890 
9.88  916 
9.88  942 
9.88  968 
9.88  994 

0.11  110 
0.11  084 
0.11  058 
0.11  032 
0.11  006 

9.89  801 
9.89  791 
9.89  781 
9.89  771 
9.89  761 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.78  772 
9.78  788 
9.78  805 
9.78  821 
9.78  837 

9.89  020 
9.89  046 
9.89  073 
9.89  099 
9.89  125 

0.10  980 
0.10  954 
0.10  927 
0.10  901 
0.10  875 

9.89  752 
9.89  742 
9.89  732 
9.89  722 
9.89  712 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.78  853 
9.78  869 
9.78  886 
9.78  902 
9.78  918 

9.89  151 
9.89  177 
9.89  203 
9.89  229 
9.89  255 

0.10  849 
0.10  823 
0.10  797 
0.10  771 
0.10  745 

9.89  702 
9.89  693 
9.89  683 
9.89  673 
9.89  663 

5 
4 
3 
2 
1 

60 

9.78  934 

9.89  281 

0.10  719 

9.89  653 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

[79] 


i 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L*  Cos* 

fJ1° 

0 

i 

2 
3 

4 

9.78  934 
9.78  950 
9.78  967 
9.78  983 
9.78  999 

9.89  281 
9.89  307 
9.89  333 
9.89  359 
9.89  385 

0.10  719 
0.10  693 
0.10  667 
0.10  641 
0.10  615 

9.89  653 
9.89  643 
9.89  633 
9.89  624 
9.89  614 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.79  015 
9.79  031 
9.79  047 
9.79  063 
9.79  079 

9.89  411 
9.89  437 
9.89  463 
9.89  489 
9.89  515  ' 

0.10  589 
0.10  563 
0.10  537 
0.10  511 
0.10  485 

9.89  604 
9.89  594 
9.89  584 
9.89  574 
9.89  564 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.79  095 
9.79  111 
9.79  128 
9.79  144 
9.79  160 

9.89  541 
9.89  567 
9.89  593 
9.89  619 
9.89  645 

0.10  459 
0.10  433 
0.10  407 
0.10  381 
0.10  355 

'  9.89  554 
9.89  544 
9.89  534 
9.89  524 
9.89  514 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.79  176 
9.79  192 
9.79  208 
9.79  224 
9.79  240 

9.89  671 
9.89  697 
9.89  723 
9.89  749 
9.89  775 

0.10  329 
0.10  303 
0.10  277 
0.10  251 
0.10  225 

9.89  504 
9.89  495 
9.89  485 
9.89  475 
9.89  465 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.79  256 
9.79  272 
9.79  288 
9.79  304 
9.79  319 

9.89  801 
9.89  827 
9.89  853 
9.89  879 
9.89  905 

0.10  199 
0.10  173 
0.10  147 
0.10  121 
0.10  095 

9.89  455 
9.89  445 
9.89  435 
9.89  425 
9.89  415 

40 

39 
38 
37 
36 

38° 

25 
26 
27 
28 
29 

9.79  335 
9.79  351 
9.79  367 
9.79  383 
9.79  399 

9.89  931 
9.89  957 
9.89  983 
9.90  009 
9.90  035 

0.10  069 
0.10  043 
0.10  017 
0.09  991 
0.09  965 

9.89  405 
9.89  395 
9.89  385 
9.89  375 
9.89  364 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.79  415 
9.79  431 
9.79  447 
9.79  463 
9.79  478 

9.90  061 
9.90  086 
9.90  112 
9.90  138 
9.90  164 

0.09  939 
0.09  914 
0.09  888 
0.09  862 
0.09  836 

9.89  354 
9.89  344 
9.89  334 
9.89  324 
9.89  314 

30 

29 
28 
27 
26 

tf  J. 

35 
36 
37 
38 
39 

9.79  494 
9.79  510 
9.79  526 
9.79  542 
9.79  558 

9.90  190 
9.90  216 
9.90  242 
9.90  268 
9.90  294 

0.09  810 
0.09  784 
0.09  758 
0.09  732 
0.09  706 

9.89  304 
9.89  294 
9.89  284 
9.89  274 
9.89  264 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.79  573 
9.79  589 
9.79  605 
9.79  621 
9.79  636 

9.90  320 
9.90  346 
9.90  371 
9.90  397 
9.90  423 

0.09  680 
0.09  654 
0.09  629 
0.09  603 
0.09  577 

9.89  254 
9.89  244 
9.89  233 
9.89  223 
9.89  213 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.79  652 
9.79  668 
9.79  684 
9.79  699 
9.79  715 

9.90  449 
9.90  475 
9.90  501 
9.90  527 
9.90  553 

0.09  551 
0.09  525 
0.09  499 
0.09  473 
0.09447 

9.89  203 
9.89  193 
9.89  183 
9.89  173 
9.89  162 

15 
14 
13 
12 
11 

60 

51 

52 
53 
54 

9.79  731 
9.79  746 
9.79  762 
9.79  778 
9.79  793 

9.90  578 
9.90  604 
9.90  630 
9.90  656 
9.90  682 

0.09  422 
0.09  396 
0.09  370 
0.09  344 
0.09  318 

9.89  152 
9.89  142 
9.89  132 
9.89  122 
9.89  112 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.79  809 
9.79  825 
9.79  840 
9.79  856 
9.79  872 

9.90  708 
9.90  734 
9.90  759 
9.90  785 
9.90  811 

0.09  292 
0.09  266 
0.09  241 
0.09  215 
0.09  189 

9.89  101 
9.89  091 
9.89  081 
9.89  071 
9.89  060 

5 
4 
3 
2 
1 

60 

9.79  887 

9.90  837 

0.09  163 

9.89  050 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

F 

[80] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

50° 

0 

i 

2 
3 
4 

9.79  887 
9.79  903 
9.79  918 
9.79  934 
9.70  950 

9.90  837 
9.90  863 
9.90  889 
9.90  914 
9.90  940 

0.09  163 
0.09  137 
0.09  111 
0.09  086 
0.09  060 

9.89  050 
9.89  040 
9.89  030 
9.89  020 
9.89  009 

60 

59 
58 
57 
56 

5 
6 

7  ' 
8 
9 

9.79  965 
9.79  981 
9.79  996 
9.80  012 
9.80  027 

9.90  966 
9.90  992 
9.91  018 
9.91  043 
9.91  069 

0.09  034 
0.09  008 
0.08  982 
0.08  957 
0.08  931 

9.88  999 
9.88  989 
9.88  978 
9.88  968 
9.88  958 

55 
54 
53 
52 
51 

. 

10 

11 
12 
13 
14 

9.80  043 
9.80  058 
9.80  074 
9.80  089 
9.80  105 

9.91  095 
9.91  121 
9.91  147 
9.91  172 
9.91  198 

0.08  905 
0.08  879 
0.08  853 
0.08  828 
0.08  802 

9.88  948 
9.88  937 
9.88  927 
9.88  917 
9.88  906 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.80  120 
9.80  136 
9.80  151 
9.80  166 
9.80  182 

9.91  224 
9.91  250 
9.91  276 
9.91  301 
9.91  327 

0.08  776 
0.08  750 
0.08  724 
0.08  699 
0.08  673 

9.88  896 
9.88  886 
9.88  875 
9.88  865 
9.88  855 

45 
44 
43 
42 
41 

39° 

20 

21 
22 
23 
24 

9.80  197 
9.80  213 
9.80  228 
9.80  244 
9.80  259 

9.91  353 
9.91  379 
9.91  404 
9.91  430 
9.91  456 

0.08  647 
0.08  621 
0.08  596 
0.08  570 
0.08  544 

9.88  844 
9.88  834 
9.88  824 
9.88  813 
9.88  803 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

9.80  274 
9.80  290 
9.80  305 
9.80  320 
9.80  336 

9.91  482 
9.91  507 
9.91  533 
9.91  559 
9.91  585 

0.08  518 
0.08  493 
0.08  467 
0.08  441 
0.08  415 

9.88  793 
9.88  782 
9.88  772 
9.88  761 
9.88  751 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.80  351 
9.80  366 
9.80  382 
9.80  397 
9.80  412 

9.91  610 
9.91  636 
9.91  662 
9.91  688 
9.91  713 

0.08  390 
0.08  364 
0.08  338 
0.08  312 
0.08  287 

9.88  741 
9.88  730 
9.88  720 
9.88  709 
9.88  699 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.80  428 
9.80  443 
9.80  458 
9.80  473 
9.80  489 

9.91  739 
9.91  765 
9.91  791 
9.91  816 
9.91  842 

0.08  261 
0.08  235 
0.08  209 
0.08  184 
0.08  158 

9.88  688 
9.88  678 
9.88  668 
9.88  657 
9.88  647 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.80  504 
9.80  519 
9.80  534 
9.80  550 
9.80  565 

9.91  868 
9.91  893 
9.91  919 
9.91  945 
9.91  971 

0.08  132 
0.08  107 
0.08  081 
0.08  055 
0.08  029 

9.88  636 
9.88  626 
9.88  615 
9.88  605 
9.88  594 

20 

19 
18 
17 
16 

45 

46 
47 
48 
49 

9.80  580 
9.80  595 
9.80  610 
9.80  625 
9.80  641 

9.91  996 
9.92  022 
9.92  048 
9.92  073 
9.92  099 

0.08  004 
0.07  978 
0.07  952 
0.07  927 
0.07  901 

9.88  584 
9.88  573 
9.88  563 
9.88  552 
9.88  542 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.80  656 
9.80  671 
9.80  686 
9.80  701 
9.80  716 

9.92  125 
9.92  150 
9.92  176 
9.92  202 
9.92  227 

0.07  875 
0.07  850 
0.07  824 
0.07  798 
0.07  773 

9.88  531 
9.88  521 
9.88  510 
9.88  499 
9.88  489 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.80  731 
9.80  746 
9.80  762 
9.80  777 
9.80  792 

9.92  253 
9.92  279 
9.92  304 
9.92  330 
9.92  356 

0.07  747 
0.07  721 
0.07  696 
0.07  670 
0.07  644 

9.88  478 
9.88  468 
9.88  457 
9.88  447 
9.88  436 

5 
4 
3 
2 
1 

60 

9.80  807 

9.92  381 

0.07  619 

9.88  425  • 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

i 

[81] 


i 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 

i 

2 
3 

4 

9.80  807 
9.80  822 
9.80  837 
9.80  852 
9.80  867 

9.92  381 
9.92  407 
9.92  433 
9.92  458 
9.92  484 

0.07  619 
0.07  593 
0.07  567 
0.07  542 
0.07  516 

9.88  425 
9.88  415 
9.88  404 
9.88  394 
9.88  383 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.80  882 
9.80  897 
9.80  912 
9.80  927 
9.80  942 

9.92  510 
9.92  535 
9.92  561 
9.92  587 
9.92  612 

0.07  490 
0.07  465 
0.07  439 
0.07  413 
0.07  388 

9.88  372 
9.88  362 
9.88  351 
9.88  340 
9.88  330 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.80  957 
9.80  972 
9.80  987 
9.81  002 
9.81  017 

9.92  638 
9.92  663 
9.92  689 
9.92  715 
9.92  740 

0.07  362 
0.07  337 
0.07  311 
0.07  285 
0.07  260 

9.88  319 
9.88  308 
9.88  298 
9.88  287 
9.88  276 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.81  032 
9.81  047 
9.81  061 
9.81  076 
9.81  091 

9.92  766 
9.92  792 
9.92  817 
9.92  843 
9.92  868 

0.07  234 
0.07  208 
0.07  183 
0.07  157 
0.07  132 

9.88  266 
9.88  255 
9.88  244 
9.88  234 
9.88  223 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.81  106 
9.81  121 
9.81  136 
9.81  151 
9.81  166 

9.92  894 
9.92  920 
9.92  945 
9.92  971 
9.92  996 

0.07  106 
0.07  080 
0.07  055 
0.07  029 
0.07  004 

9.88  212 
9.88  201 
9.88  191 
9.88  180 
9.88  169 

40 

39 
38 
37 
36 

40° 

25 
26 
27 
28 
29 

9.81  180 
9.81  195 
9.81  210 
9.81  225 
9.81  240 

9.93  022 
9.93  048 
9.93  073 
9.93  099 
9.93  124 

0.06  978 
0.06  952 
0.06  927 
0.06  901 
0.06  876 

9.88  158 
9.88  148 
9.88  137 
9.88  126 
9.88  115 

35 
34 
33 
32 
31 

49 

30 

31 
32 
33 
34 

9.81  254 
9.81  269 
9.81  284 
9.81  299 
9.81  314 

9.93  150 
9.93  175 
9.93  201 
9.93  227 
9.93  252 

0.06  850 
0.06  825 
0.06  799 
0.06  773 
0.06  748 

9.88  105 
9.88  094 
9.88  083 
9.88  072 
9.88  061 

30 

29 

28 
27 
26 

35 
36 
37 
38 
39 

9.81  328 
9.81  343 
9.81  358 
9.81  372 
9.81  387 

9.93  278 
9.93  303 
9.93  329 
9.93  354 
9.93  380 

0.06  722 
0.06  697 
0.06  671 
0.06  646 
0.06  620 

9.88  051 
9.88  040 
9.88  029 
9.88  018 
9.88  007 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.81  402 
9.81  417 
9.81  431 
9.81  446 
9.81  461 

9.93  406 
9.93  431 
9.93  457 
9.93  482 
9.93  508 

0.06  594 
0.06  569 
0.06  543 
0.06  518 
0.06  492 

9.87  996 
9.87  985 
9.87  975 
9.87  964 
9.87  953 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.81  475 
9.81  490 
9.81  505 
9.81  519 
9.81  534 

9.93  533 
9.93  559 
9.93  584 
9.93  610 
9.93  636 

0.06  467 
0.06  441 
0  06  416 
0.06  390 
0.06  364 

9.87  942 
9.87  931 
9.87  920 
9.87  909 
9.87  898 

15 
14 
13 
12 
11 

50. 

51 
52 
53 
54 

9.81  549 
9.81  563 
9.81  578 
9.81  592 
9.81  607 

9.93  661 
9.93  687 
9.93  712 
9.93  738 
9.93  763 

0.06  339 
0.06  313 
0.06  288 
0.06  262 
0.06  237 

9.87  887 
9.87  877 
9.87  866 
9«.87  855 
9.87  844 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.81  622 
9.81  636 
9.81  651 
9.81  665 
9.81  680 

9.93  789 
9.93  814 
9.93  840 
9.93  865 
9.93  891 

0.06  211 
0.06  186 
0.06  160 
0.06  135 
0.06  109 

9.87-833 
9.87  822 
9.87  811 
9.87  800 
9.87  789 

5 
4 
3 
2 
1 

60 

9.81  694 

9.93  916 

0.06  084 

9  87  778 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[82] 


I 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

48° 

0 

1 

2 
3 
4 

9.81  694 
9.81  709 
9.81  723 
9.81  738 
9.81  752 

9.93  916 
9.93  942 
9.93  967 
9.93  993 
9.94  018 

0.06  084 
0.06  058 
0.06  033 
0.06  007 
0.05  982 

9.87  778 
9.87  767 
9.87  756 
9.87  745 
9.87  734 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.81  767 
9.81  781 
9.81  796 
9.81  810 
9.81  825 

9.94  044 
9.94  069 
9.94  095 
9.94  120 
9.94  146 

0.05  956 
0.05  931 
0.05  905 
0.05  880 
0.05  854 

9.87  723 
9.87  712 
9.87  701 
9.87  630 
9.87  679 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.81  839 
9.81  854 
9.81  868 
9.81  882 
9.81  897 

9.94  171 
9.94  197 
9.94  222 
9.94  248 
9.94  273 

0.05  829 
0.05  803 
0.05  778 
0.05  752 
0.05  727 

9.87  668 
9.87  657 
9.87  646 
9.87  635 
9.87  624 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.81  911 
9.81  926 
9.81  940 
9.81  955 
9.81  969 

9.94  299 
9.94  324 
9.94  350 
9.94  375 
9.94  401 

0.05  701 
0.05  676 
0.05  650 
0.05  625 
0.05  599 

9.87  613 
9.87  601 
9.87  590 
9.87  579 
9.87  568 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.81  983 
9.81  998 
9.82  012 
9.82  026 
9.82  041 

9.94  426 
9.94  452 
9.94  477 
9.94  503 
9.94  528 

0.05  574 
0.05  548 
0.05  523 
0.05  497 
0.05  472 

9.87  557 
9.87  546 
9.87  535 
9.87  524 
9.87  513 

40 

39 
38 
37 
36 

41 

25 
26 
27 
28 
29 

9.82  055 
9.82  069 
9.82  084 
9.82  098 
9.82  112 

9.94  554 
9.94  579 
9.94  604 
9.94  630 
9.94  655 

0.05  446 
0.05  421 
0.05  396 
0.05  370 
0.05  345 

9.87  501 
9.87  490 
9.87  479 
9.87  468 
9.87  457 

35 
34 

33 
32 
31 

30 

31 
32 
33 

34 

9.82  126 
9.82  141 
9.82  155 
9.82  169 
9.82  184 

9.94  681 
9.94  706 
9.94  732 
9.94  757 
9.94  783 

0.05  319 
0.05  294 
0.05  268 
0.05  243 
0.05  217 

9.87  446 
9.87  434 
9.87  423 
9.87  412 
'  9.87401 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.82  198 
9.82  212 
9.82  226 
9.82  240 
9.82  255 

9.94  808 
9.94  834 
9.94  859 
9.94  884 
9.94  910 

0.05  192 
0.05  166 
O.*05  141 
0.05  116 
0.05  090 

9.87  390 
9.87  378 
9.87  367 
9.87  356 
9.87  345 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.82  269 
9.82  283 
9.82  297 
9.82  311 
9.82  326 

9.94  935 
9.94  961 
9.94  986 
9.95  012 
9.95  037 

0.05  065 
0.05  039 
0.05  014 
0.04  988 
0.04  963 

9.87  334 
9.87  322 
9.87  311 
9.87  300 
9.87  288 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.82  340 
9.82  354 
9.82  368 
9.82  382 
9.82  396 

9.95  062 
9.95  088 
9.95  113 
9.95  139 
9.95  164 

0.04  938 
0.04  912 
0.04  887 
0.04  861 
0.04  836 

9.87  277 
9.87  266 
9.87  255 
9.87  243 
9.87  232 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.82  410 
§.82  424 
9.82  439 
9.82  453 
9.82  467 

9.95  190 
9.95  215 
9.95  240 
9.95  266 
9.95  291 

0.04  810 
0.04  785 
0.04  760 
0.04  734 
0.04  709 

9.87  221 
9.87  209 
9.87  198 
9.87  187 
9.87  175 

10 

9 
8 
7 

6 

55 
56 
57 
58 
59 

9.82  481 
9.82  495 
9.82  509 
9.82  523 
9.82  537 

9.95  317 
9.95  342 
9.95  368 
9.95  393 
9.95  418 

0.04  683 
0.04  658 
0.04  632 
0.04  607 
0.04  582 

9.87  164 
9.87  153 
9.87  141 
9.87  130 
9.87  119 

5 
4 
3 
2 
1 

60 

9.82  551 

9.95  444 

0.04  556 

9.87  107 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

[83] 


r 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

0 
1 

2 
3 
4 

9.82  551 
9.82  565 
9.82  579 
9.82  593 
9.82  607 

9.95  444 
9.95  469 
9.95  495 
9.95  520 
9.95  545 

0.04  556 
0.04  531 
0.04  505 
0.04  480 
0.04  455 

9.87  107 
9.87  096 
9.87  085 
9.87  073 
9.87  062 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.82  621 
9.82  635 
9.82  649 
9.82  663 
9.82  677 

9.95  571 
9.95  596 
9.95  622 
9.95  647 
9.95  672 

0.04  429 
0.04  404 
0.04  378 
0.04  353 
0.04  328 

9.87  050 
9.87  039 
9.87  028 
9.87  016 
9.87  005 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.82  691 
9.82  705 
9.82  719 
9.82  733 
9.82  747 

9.95  698 
9.95  723 
9.95  748 
9.95  774 
9.95  799 

0.04  302 
0.04  277 
0.04  252 
0.04  226 
0.04  201 

9.86  993 
9.86  982 
9.86  970 
9.86  959 
9.86  947 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.82  761 
9.82  775 
9.82  788 
9.82  802 
9.82  816 

9.95  825 
9.95  850 
9.95  875 
9.95  901 
9.95  926 

0.04  175 
0.04  150 
0.04  125 
0.04  099 
0.04  074 

9.86  936 
9.86  924 
9.86  913 
9.86  902 
9.86  890 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.82  830 
9.82  844 
9.82  858 
9.82  872 
9.82  885 

9.95  952 
9.95  977 
9.96  002 
9.96  028 
9.96  053 

0.04  048 
0.04  023 
0.03  998 
0.03  972 
0.03  947 

9.86  879 
9.86  867 
9.86  855 
9.86  844 
9.86  832 

40 

39 
38 
37 
36 

42C 

25 
26 
27 
28 
29 

9.82  899 
9.82  913 
9.82  927 
9.82  941 
9.82  955 

9.96  078 
9.96  104 
9.96  129 
9.96  155 
9.96  180 

0.03  922 
0.03  896 
0.03  871 
0.03  845 
0.03  820 

9.86  821 
9.86  809 
9.86  798 
9.86  786 
9.86  775 

35 
34 
33 
32 
31 

47° 

30 

31 
32 
33 
34 

9.82  968 
9.82  982 
9.82  996 
9.83  010 
9.83  023 

9.96  205 
9.96  231 
9.96  256 
9.96  281 
9.96  307 

0.03  795 
0.03  769 
0.03  744 
0.03  719 
0.03  693 

9.86  763 
9.86  752 
9.86  740 
9.86  728 
9.86  717 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.83  037 
9.83  051 
9.83  065 
9.83  078 
9.83  092 

9.96  332 
9.96  357 
9.96  383 
9.96  408 
9.96  433 

0.03  668 
0.03  643 
0.03  617 
0.03  592 
0.03  567 

9.86  705 
9.86  694 
9.86  682 
9.86  670 
9.86  659 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.83  106 
9.83  120 
9.83  133 
9.83  147 
9.83  161 

9.96  459 
9.96  484 
9.96  510 
9.96  535 
9.96  560 

0.03  541 
0.03  516 
0.03  490 
.0.03  465 
0.03  440 

9.86  647 
9.86  635 
9.86  624 
9.86  612 
9.86  600 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.83  174 
9.83  188 
9.83  202 
9.83  215 
9.83  229 

9.96  586 
9.96  611 
9.96  636 
9.96  662 
9.96  687 

0.03  414 
0.03  389 
0.03  364 
0.03  338 
0.03  313 

9.86  589 
9.86  577 
9.86  565 
9.86  554 
9.86  542 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.83  242 
9.83  256 
9.83  270 
9.83  283 
9.83  297 

9.96  712 
9.96  738 
9.96  763 
9.96  788 
9.96  814 

0.03  288 
0.03  262 
0.03  237 
0.03  212 
0.03  186 

9.86  530 
9.86  518 
9.86  507 
9.86  495 
9.86  483 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.83  310 
9.83  324 
9.83  338 
9.83  351 
9.83  365 

9.96  839 
9.96  864 
9.96  890 
9.96  915 
9.96  940 

0.03  161 
0.03  136 
0.03  110 
0.03  085 
0.03  060 

9-86472 
9.86  460 
9.86  448 
9.86  436 
9.86  425 

5 
4 
3 
2 
1 

60 

9.83  378 

9.96  966 

0.03  034 

9.86  413 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

/ 

t 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

46° 

0 

i 

2 
3 
4 

9.83  378 
9.83  392 
9.83  405 
9.83  419 
9.83  432 

9.96  966 
9.96  991 
9.97  016 
9.97  042 
9.97  067 

0.03  034 
0.03  009 
0.02  984 
0.02  958 
0.02  933 

9.86  413 
9.86  401 
9.86  389 
9.86  377 
9.86  366 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.83  446 
9.83  459 
9.83  473 
9.83  486 
9.83  500 

9.97  092 
9.97  118 
9.97  143 
9.97  168 
9.97  193 

0.02  908 
0.02  882 
0.02  857 
0.02  832 
0.02  807 

9.86  354 
9.86  342 
9.86  330 
9.86  318 
9.86  306 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.83  513 
9.83  527 
9.83  540 
9.83  554 
9.83  567 

9.97  219 
9.97  244 
9.97  269 
9.97  295 
9.97  320 

'0.02  781 
0.02  756 
0.02  731 
0.02  705 
0.02  680 

9.86  295 
9.86  283 
9.86  271 
9.86  259 
9.86  247 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.83  581 
9.83  594 
9.83  608 
9.83  621 
9.83  634 

9.97  345 
9.97  371 
9.97  396 
9.97  421 
9.97  447 

0.02  655 
0.02  629 
0.02  604 
0.02  579 
0.02  553 

9.86  235 
9.86  223 
9.86  211 
9.86  200 
9.86  188 

45 
44 
43 
42 
41 

43° 

20 

21 
22 
23 
24 

9.83  648 
9.83  661 
9.83  674 
9.83  688 
9.83  701 

9.97  472 
9.97  497 
9.97  523 
9.97  548 
9.97  573 

0.02  528 
0.02  503 
0.02  477 
0.02  452 
0.02  427 

9.86  176 
9.36  164  . 
9.86  152 
9.86  140 
9.86  128 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

9.83  715 
9.83  728 
9.83  741 
9.83  755 
9.83  768 

9.97  598 
9.97  624 
9.97  649 
9.97  674 
9.97  700 

0.02  402 
0.02  376 
0.02  351 
0.02  326 
0.02  300 

9.86  116 
9.8G  104 
9.83  092 
9.86  080 
9.86  068 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.83  781 
9.83  595 
9.83  808 
9.83  821 
9.83  834 

9.97  725 
9.97  750 
9.97  776 
9.97  801 
9.97  826 

0.02  275 
0.02  250 
0.02  224 
0.02  199 
0.02  174 

9.8G  056 
9.8G  044 
9.86  032 
9.86  020 
9.86  008 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.83  848 
9.83  861 
9.83  874 
9.83  887 
9.83  901 

9.97  851 
9£7  877 
9.97  902 
9.97  927 
9.97  953 

0.02  149 
0.02  123 
0.02  098 
0.02  073 
0.02  047 

9.85  996 
9.85  984 
9.85  972 
9.85  960 
9.85  948 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.83  914 
9.83  927 
9.83  940 
9.83  954 
9.83  967 

9.97  978 
'  9.98  003 
9.98  029 
9.98  054 
9.98  079 

0.02  022 
0.01  997 
0.01  971 
0.01  946 
0.01  921 

9.85  936 
9.85  924 
9.85  912 
9.85  900 
9.85  888 

20 

19 
18 
17 

16 

45 

46 
47 
48 
49 

9.83  980 
9.83  993 
9.84  006 
9.84  020 
9.84  033 

9.98  104 
9.98  130 
9.98  155 
9.98  180 
9.98  206 

0.01  896 
0.01  870 
0.01  845 
0.01  820 
0.01  794 

9.85  876 
9.85  864 
9.85  851 
9.85  839 
9.85  827 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.84-  046 
9.84  059 
9.84  072 
9.84  085 
9.84  098 

9.98  231 
9.98  256 
9.98  281 
9.98  307 
9.98  332 

0.01  769 
0.01  744 
0.01  719 
0.01  693 
0.01  668 

9.85  815 
9.85  803 
9.85  791 
9.85  779 
9.85  766 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

9.84  112 
9.84  125 
9.84  138 
9.84  151 
9.84  164 

9.98  357 
9.98  383 
9.98  408 
9.98  433 
9.98  458 

0.01  643 
0.00  617 
0.01  592 
0.01  567 
0.01  542 

9.85  754 
9.85  742 
9.85  730 
9.85  718 
9.85  706 

5 
4 
3 
2 
1 

60 

9.84  177 

9.98  484 

0.01  516 

9.85  693 

0 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.  Sin. 

f 

[85] 


/ 

L.  Sin. 

L.  Tang. 

L.  Cotg. 

L.  Cos. 

45° 

0 

i 

2 
3 
4 

9.84  177 
9.84  190 
9.84  203 
9.84  216 
9.84  229 

9.98  484 
9.98  509 
9.98  534 
9.98  560 
9.98  585 

0.01  516 
0.01  491 
0.01  466 
0.01  440 
0.01  415 

9.85  693 
9.85  681 
9.85  669 
9.85  657 
9.85  645 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

9.84  242 
9.84  255 
9.84  269 
9.84  282 
9.84  295 

9.98  610 
9.98  635 
9.98  661 
9.98  686 
9.98  711 

0.01  390 
0.01  365 
0.01  339 
0.01  314 
0.01  289 

9.85  632 
9.85  620 
9.85  608 
9.85  596 
9.85  583 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

9.84  308 
9.84  321 
9.84  334 
9.84  347 
9.84  360 

9.98  737' 
9.98  762 
9.98  787 
9.98  812 
9.98  838 

0.01  263 
0.01  238 
0.01  213 
0.01  188 
0.01  162 

9.85  571 
9.85  559 
9.85  547 
9.85  534 
9.85  522 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

9.84  373 
9.84  385 
9.84  398 
9.84  411 
9.84  424 

9.98  863 
9.98  888 
9.98  913 
9.98  939 
9.98  964 

0.01  137 
0.01  112 
0.01  087 
0.01  061 
0.01  036 

9.85  510 
1  9.85  497 
9.85  485 
9.85  473 
9.85  460 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

9.84  437 
9.84  450  . 
9.84  463 
9.84  476 
9.84  489 

9.98  989 
9.99  015 
9.99  040 
9.99  065 
9.99  090 

0.01  Oil 
0.00  985 
0.00  960 
0.00  935 
0.00  910 

9.85  448 
9.85436 
9.85  423 
9.85  411 
9.85  399 

40 

39 
38 
37 
36 

44° 

25 
26 
27 
28 
29 

9.84  502 
9.84  515 
9.84  528 
9.84  540 
9.84  553 

9.99  116 
9.99  141 
9.99  166 
9.99  191 
9.99  217 

0.00  884 
0.00  859 
0.00  834 
0.00  809 
0.00  783 

9.85  386 
9.85  374 
9.85  361 
9.85  349 
9.85  337 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

9.84  566 
9.84  579 
9.84  592 
9.84  605 
9.84  618 

9.99  242 
9.99  267 
9.99  293 
9.99  318 
9.99  343 

0.00  758 
0.00  733 
0.00  707 
0.00  682 
0.00  657 

9.85  324 
9.85  312 
9.85  299 
9.85  287 
9.85  274 

30 

29 
28 
27 
26 

35 
36- 
37 
38 
39 

9.84  630 
9.84  643 
9.84  656 
9.84  669 
9.84  682 

9.99  368 
9.99  394 
9.99  419 
9.99  444 
9.99  469 

0.00  632 
0.00  606 
0.00  581 
0.00  556 
0.00  531 

9.85  262 
9.85'250 
9.85  237 
9.85  225 
9.85  212 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

9.84  694 
9.84  707 
9.84  720 
9.84  733 
9.84  745 

9.99  495 
9.99  520 
9.99  545 
9.99  570 
9.99  596 

0.00  505 
0.00  480 
0.00  455 
0.00  430 
0.00  404 

9.85  200 
9.85  187 
9.85  175 
9.85  162 
9.85  150 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

9.84  758 
9.84  771 
9.84  784 
9.84  796 
9.84  809 

9.99  621 
9.99  646 
9.99  672 
9.99  697 
9.99  722 

0.00  379 
0.00  354 
0.00  328 
0.00  303 
0.00  278 

9.85  137 
9.85  125 
9.85  112 
9.85  100 
9.85  087 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

9.84  822 
9.84  835 
9.84  847 
9.84  860 
9.84  873 

9.99  747 
9.99  773 
9.99  798 
9.99  823 
9.99  848 

0.00  253 
0.00  227 
0.00  202 
0.00  177 
0.00  152 

9.85-074 
9.85  062 
9.85  049 
9.85  037 
9.85  024 

10 

9 
8 
7 
6  - 

55 
56 
57 
58 
59 

9.84  885 
9.84  898 
9.84  911 
9.84  923 
9.84  936 

9.99  874 
9.99  899 
9.99  924 
9.99  949 
9.99  975 

0.00  126 
0.00  101 
0.00  076 
0.00  051 
0.00  025 

9.85  012 
9.84  999 
9.84  986 
9.84  974 
9.84  961 

5 
.  4 
3 
2 
1 

60 

9.84  949 

0.00  000 

0.00  000 

9.84  949 

0 

L.  Cos. 

L,  Cotg. 

L.  Tang. 

L.  Sin. 

; 

[86] 


TABLE  IV 


AUXILIARY  FIVE-PLACE  TABLE 


FOR 


SMALL  ANGLES 


[87] 


// 

i 

8 

T 

w 

T' 

L.  Sin. 

0 
60 
120 
180 
240 

0 
1 
2 
3 
4 

4.68557 
.68557 
.68557 
.68557 
.68557 

4.68557 
.68557 
.68557 
.68557 
.68558 

5.31443 
.31443 
.31443 
.31443 
.31443 

5.31443 
.31443 
.31443 
.31443 
.31442 

6.46373 
.76476 
.94085 
7.06579 

300 
360 
420 
480 
540 

5 
6 
7 
8 
9 

4.68557 
.68557 
.68557 
.68557 
.68557 

4.68558 
.68558 
.68558 
.68558 
.68558 

5.31443 
.31443 
.31443 
.31443 
.31443 

5.31442 
.31442 
.31442 
.31442 
.31442 

7.16270 
.24188 
.30882 
.36682 
.41797 

600 
660 
720 
780 
840 

10 
11 
12 
13 
14 

4.68557 
.68557 
.68557 
.68557 
.68557" 

4.68558 
.68558 
.68558 
.68558 
.68558 

5.31443 
.31443 
.31443 
.31443 
.31443 

5.31442 
.31442 
.31442 
.31442 
.31442 

7.46373 
.50512 
.54291 
.57767 
.60985 

0° 

900 
960 
1020 
1080 
1140 

15 
16 
17 
18 
19 

4.68557 
.68557 
.68557 
.68557 
.68557 

4.68558 
.68558 
.68558 
.68558 
.68558 

5.31443 
.31443 
.31443 
.31443 
.31443 

5.31442 
.31442 
.31442 
.31442 
.31442 

7.63982 
.66784 
.69417 
.71900 
.74248 

1200 
1260 
1320 
1380 
1440 

20 
21 
22 
23 
24 

4.68557 
.6-8557 
.68557 
.68557 
.68557 

4.68558 
.68558 
.68558 
.68558 
.68558 

5.31443 
^.31443 
.31443 
.31443 
.31443 

5.31442 
.31442 
.31442 
.31442 
.31442 

7.76475 
.78594 
.80615 
.82545 
.84393 

1500 
1560 
1620 
1680 
1740 

25 
26 
27 
28 
29 

4.68557 
.68557 
.68557 
.68557 
.68557 

4.68558 
.68558 
.68558 
.68558 
.68559 

5.31443 
.31443 
.31443 
.31443 
.31443 

5.31442 
.31442 
.31442 
.31442 
.31441 

7.86166 
.87870 
.89509 
.91088 
.92612 

1800 
1860 
1920 
1980 
2040 

30 
31 
32 
33 
34 

4.68557 
.68557 
.68557 
.68557 
.68557 

4.68559 
.68559 
.68559 
.68559 
.68559 

5.31443 
.31443 
.31443 
.31443 
.31443 

5.31441 
.31441 
.31441 
.31441 
.31441 

7.94084 
.95508 
.96887 
.98223 
.99520 

2100 
2160 
2220 
2280 
2340 

35 
36 
37 
38 
39 

4.68557 
.68557 
.68557 
.68557 
.68557 

4.68559 
.68559 
.68559 
.68559 
.68559 

5.31443 
.31443 
.31443 
.31443 
.31443 

5.31441 
.31441 
.31441 
.31441 
.31441 

8.00779 
.02002 
.03192 
.04350 
.05478 

2400 
2460 
2520 
2580 
2640 

40 
41 

42 
43 
44 

4.68557 
.68556 
.68556 
.68556 
.68556 

4.68559 
.68560 
.68560 
.68560 
.68560 

5.31443 
.31444 
.31444 
.31444 
".31444 

5.31441 
.31440 
.31440 
.31440 
.31440 

8.06578 
.07650 
.08696 
.09718 
.10717 

2700 
2760 
2820 
2880 
2940 

45 
46 
47 
48 
49 

4.68556 
.68556 
.68556 
.68556 
.68556 

4.68560 
.68560 
.68560 
.68560 
.68560 

f!444 
1444 
.31444 
.31444 
.31444 

.  5.31440 
.31440 
.31440 
.31440 
.31440 

8.11693 
.12647 
.13581 
.14495 
.15391 

3000 
3060 
3120 
3180 
3240 

50 
51 
52 
53 
54 

4.68556 
.68556 
.68556 
.68556 
.68556 

4.68561 
.68561 
.68561 
.68561 
.68561 

5.31444 
.31444 
.31444 
.31444 
.31444 

5.31439 
.31439 
.31439 
.31439 
.31439 

8.16268 
.17128 
.17971 
.18798 
.19610 

3300 
3360 
3420 
3480 
3540 

55 
56 
57 
58 
.59 

4.68556 
.68556 
.68555 
.68555 
.68555 

4.68561 
.68561 
.68561 
.68562 
.68562 

5.31444 
.31444 
.31445 
.31445 
.31445 

5.31439 
.31439 
.31439 
.31438 
.31438 

8.20407 
.21189 
.21958 
.22713 
.23456 

3600 

60 

4.68555 

4.68562 

5.31445 

5.31438 

8.24186 

[88] 


// 

/ 

S 

T 

S' 

T' 

L.  Sin. 

3600 
3660 
3720 
3780 
3840 

0 
1 
2 
3 

4 

4.68555 
.68555 
.68555 
.68555 
.68555 

4.68562 
.68562 
.68562 
.68562 
.68663 

5.31445 
.31445 
.31445 
.31445 
.31445 

5.31438 
.31438 
.31438 
.31438 
.31437 

8.24186 
.24903 
.25609 
.26304 
.26988 

3900 
3960 
4020 
4080 
4140 

5 
6 
7 
8 
9 

4.68555 
.68555 
.68555 
.68555 
.68555 

4.68563 
.68563 
.68563 
.68563 
.68563 

5.31445 
.31445 
.31445 
.31445 
.31445 

5.31437 
.31437 
.31437 
.31437 
.31437 

8.27661 
.28324 
.28977 
.29621 
.30255 

4200 
4260 
4320 
4380 
4440 

10 
11 
12 
13 
14 

4.68554 
.68554 
.68554 
.68554 
.68554 

4.68563 
.68564 
.68564 
.68564 
.68564 

5.31446 
.31446 
.31446 
.31446 
.31446 

5.31437 
.31436 
.31436 
.31436 
.31436 

8.30879 
.31495 
.32103 
.32702 
.33292 

4500 
4560 
4620 
4680 
4740 

15 
16 
17 
18 
19 

4.68554 
.68554 
.68554 
.68554 
.68554 

4.68564 
.68565 
.68565 
.68565 
.68565 

5.31446 
.31446 
.31446 
.31446 
.31446 

5.31436 
.31435 
.31435 
.31435 
.31435 

8.33875 
.34450 
.35018 
.35578  ' 
.36131 

1° 

4800 
4860 
4920 
4980 
5040 

20 
21 
22 
23 
24^ 

4.68554 
.68553 
.68553 
.68553 
.68553 

4.68565 
.68566 
.68566 
.68566 
.68566- 

5.31446 
.31447 
.31447 
.31447 
.31447 

5.31435 
.31434 
.31434 
.31434 
.31434 

8.36678 
.37217 
.37750 
.38276 
.38794 

5100 
5160 
5220 
5280 
5340 

25- 
26, 
27 
28 
29 

4.68553 
.68553. 
.685984 
.68553 
.68553 

4.68566 
.68567 
.68567 
.68567 
.68567 

5.31447- 
.31447 
.31447 
.31447 
.31447 

5.31434" 
.31433 
.31433 
.31433 
.31433 

8.393,trf 
.39818 
.40320 
.40816 
.41307 

5400 
5460 
5520 
5580 
5640 

30 
31 
32 
33 
34 

4.68553 
.68552 
.68552 
.68552 
.68552 

4.68567 
.68568 
.68568 
.68568 
.68568 

5.31447 
.31448 
.31448 
.31448 
.31448 

5.31433 
.31432 
.31432 
.31432 
.31432 

8.41792 
.42272 
.42746 
.43216 
.43680 

5700 
5760 
5820 
5880 
5940 

35 
36 
37 
38 
39 

4.68552 
.68552 
.68552 
.68552 
.68551 

4.68569 
.68569 
.68569 
.68569 
.68569 

5.31448 
.31448 
.31448 
.31448 
.31449 

5.31431 
.31431 
.31431 
.31431 
.31431 

8.44139 
.44594 
.45044 
.45489 
.45930 

6000 
6060 
6120 
6180 
6240 

40 
41 
42 
43 
44 

4.68551 
.68551 
.68551 
.68551 
.68551 

4.68570 
.68570 
.68570 
.68570 
.68571 

5.31449 
.31449 
.31449 
.31449 
..31449 

5.31430 
.31430 
.31430 
.31430 
.31429 

8.46366 
.46799 
.47226 
.47650 
.48069 

6300  • 
6360 
6420 
6480 
6540 

45 

46 
47 
48 
49 

4.68551 
.68551 
.68550 
.68550 
.68550 

4.68571 
.68571 
.68572 
.68572 
.68572 

5.31449 
.31449 
.31450 
.31450 
.31450 

5.31429 
.31429 
.31428 
.31428 
.31428 

8.48485 
.48896 
.49304 
.49708 
.50108 

6600 
6660 
6720 
6780 
6840 

50 
51 
52 
53 
54 

4.68550 
.68550 
.68550 
.68550 
.68550 

4.68572 
.68573 
.68573 
.68573 
.68573 

5.31450 
.31450 
.31450 
.31450 
.31450 

5.31428 
.31427 
.31427 
.31427 
.31427 

8.50504 
.50897 
.51287 
.51673 
.52055 

6900 
6960 
7020 
7080 
7140 

55 
56 
57 
58 
59 

4.68549 
.68549 
.68549 
.68549 
.68549 

4.68574 
.68574 
.68574 
.68575 
.68575 

5.31451 
.31451 
.31451 
.31451 
.31451 

5.31426 
.31426 
.31426 
.31425 
.31425 

8.52434 
.52810 
.53183 
.53552 
.53919 

7200 

60 

4.68549 

4.68575 

5.31451 

5.31425 

8.54282 

[89] 


TABLE   V 


FOUR-PLACE  TABLE 


OF    THE 


NATURAL   SINE,   COSINE,   TANGENT,   AND 
•     COTANGENT 


EVERY   10'  OF   THE   QUADRANT 


[91] 


o    1 

N.  Sin. 

N.  Tan. 

N.  Cot. 

N.  Cos. 

0  00 
10 
20 
30 
40 
50 

.0000 
.0029 
.0058 
.0087 
.0116 
.0145 

.0000 
.0029 
.0058 
.0087 
.0116 
.0145 

oo 
343.77 
171.89 
114.59 
85.940 
68.750 

1.0000 
1.0000 
1.0000 
1.0000 
.9999 
.9999 

00  90 

50 
40 
30 
20  . 
-.10 

1  00 
10 
20 
^30 
s4J3 
50 

.0175 
.0204 
.0233 
Tff262 
,£291 
.0320 

.0175 
.0204 
.0233 
.0262 
.0291 
.0320 

57.290 
49.104 
42.964 
38.188 
34.368 
31.242 

.9998 
.9998 
.9997 
.9997 
.9996 
.9995 

00  89 

50 
40 
30 
20 
10 

2  00 

10 
20 
30 
40 
50 

.0349 
.0378 
.0407 
.0436 
.0465 
.0494 

.0349 
.0378 
.0407 
.0437 
.0466 
.0495 

28.636 
26.432 
24.542 
22.904 
21.470 
20.206 

.9994 
.9993 
.9992 
.9990 
.9989 
.9988 

00  88 

50 
40 
30 
20 
10 

3  00 

10 
20 
30 
40 
50 

.0523 
.0552 
.0581 
.06,10 
.0640 
.0669 

.0524 
.0553 
.0582 
.0612 
.0641 
.0670 

19.081 
18.075 
17.169 
16.350 
15.605 
14.924 

.9986 
.9985 
.9983 
.9981 
.9980 
.9978 

00  87 

50 
40 
30 
20 
10 

4  00 
10 
20 
30 
40 
50 

.0698 
.0727 
.0756 
.0785 
.0814 
.0843 

.0699 
.0729   • 
.0758 
.0787 
.0816 
0846 

14.301 
13.727 
13.197 
12.706 
12.251 
11.826 

.9976 
.9974 
.9971 
.9969 
.9967 
.9964 

00  86 

50 
40 
30 
20 
10 

5  00 

10 
20 
30 
40 
50 

.0872 
.0901 
.0929 
.0958 
.0987 
.1016 

.0875 
.0904 
.0934 
.0963 
.0992 
.1022 

11.430 
11.059 
10.712 
10.385 
10.078 
9.7882 

.9962 
.9959 
.9957 
.9954 
.9951 
.9948 

00  85 

50 
40 
30 
20 
10 

6  00 

10 
20 
30 
40 
50 

.1045 
.1074 
.1103 
.1132 
.1161 
.1190 

.1051 
.1080 
.1110 
.1139 
.1169 
.1198 

9.5144 
9.2553 
9.0098 
8.7769 
8.5555 
8.3450 

.9945 
.9942 
.9939 
.9936 
.9932 
.9929 

00  84 

50. 
40 
30 
20 
10 

7  00 
10 
20 
30 
40 
50 

.1219 
.1248 
.1276 
.1305 
.1334 
.1363 

.1228 
.1257 
.1287 
.1317 
.1346 
.1376 

8.1443' 
7.9530 
7.7704 
7.5958 
7.4287 
7.2687 

.9925 
.9922 
.9918 
.9914 
.9911 
.9907 

00  83 

50 
40 
30 
20 
10 

8  00 

10 
20 
30 
40 
50 

.1392 
.1421 
.1449 
.1478 
.1507 
.1536 

.1405 
.1435 
.1465 
.1495 
.1524 
.1554 

7.1154 
6.9682 
6.8269 
6.6912 
6.5606 
6.4348 

.99j03 
.9899 
.9^4 
.9890 
.9886 
.9881 

00  82 

50 
40 
30 
20 
10 

9  00 

.1564 

.1584 

6.3138 

.9877 

00  81 

N.  Cos.' 

N.  Cot. 

If.  Tan. 

N.Sin. 

'  r   o 

[92] 


o   ; 

N.  Sin. 

N.  Tan. 

N.  Cot. 

N.  Cos. 

9  00' 
10 
20 
30 
40 
50 

.1564 
.1593 
.1622 
.1650 
.1679 
.1708 

.1584 
.1614 
.1644 
.1673 
.1703 
.1733 

6.3138 
6.1970 
6.0844 
5.9758 
5.8708 
5.7694 

.9877 
.9872 
.9868 
.9863 
.9858 
.9853 

00  81 

50 
40 
30 
20 
10 

10  00 

10 
20 
30 
40 
50 

.1736 
.1765 
.1794  . 
.1822  / 
.1851 
.1880 

.1763 
.1793 
.1823 
.1853 
.1883 
.1914 

5.6713 
'  5.5764 
5.4845 
5.3995 
5.3093 
5.2257 

.9848 
.9843 
.9838 
.9833 
.9827 
.9822 

00  80 

50 
40 
30 
20 
10 

11  00 

10 
20 
30 
40 
50 

.1908 
.1937 
.1965 
.1994 
.2022 
.2051 

.1944 
.1974 
.2004 
.2035 
.2065 
.2095 

5.1446 
5.0658 
4.9894 
4.9152 
4.8430 
4.7729 

.9816 
.9811 
.9805 
.9799 
.9793  . 
.9787 

00  79 

50 
40 
30 
20 
10 

12  00 

10  . 
20 

30 
40 
50 

.2079 
.2108 
.2136 
.2164 
.2193 
.2221 

.2126 
.2156 
.2186 
.2217 
.2247 
.2278 

4.7046 
4.6382 
4.5736 
4.5107 
4.4494 
4.3897 

.9781 
.9775 
.9769 
.9763 
.9757 
.9750 

00  78 

0 
40 
30 
20 
10 

13  00 

10 
20 
30 
40 
50 

.2250 
.2278 
.2306 
.2334 
.2363 
.2391 

.2309 
.2339 
.2370 
.2401 
.2432 
.2462 

4.3315 
4.2747 
4.2193 
4.1653 
4.1126 
4.0611 

.9744 
.9737 
.9730 
.9724 
.9717 
.9710 

00  77 
50 
40 
30 
20 

10  -. 

14  00 

10 
20 
30 
40 
50 

.2419 
.2447 
.2476 
.2504 
.2532 
.2560 

.2493 
.2524 
.2555 
.2586 
.2617 
.2648 

4.0108 
3.9617 
3.9136 
3.8667 
3.8208 
3.7760 

.9703 
.9696- 
.9689 
.9681 
.9674 
.9667 

00  76 

50 
40 
30 
20 
10 

15  00 
10 

20 
30 
40 
50 

.2588 
.2616 
.2644 
.2672 
.2700 
.2728 

.2679  ^ 
.2711 
.2742 
.2773 
.2805 
.2836 

3.7321 
3.6891. 
3.6470 
3.6059 
3.5656 
3.5261 

.9659  - 
.9652 
.9644 
.9636 
.9628 
.9621 

00  75 

50 
40 
30 
20 
10 

16  00 

10 
20 
30 
40 
50 

.2756 
.2784 
.2812 
.2840 
.2868 
.2896 

.2867 
.2899 
.2931 
.2962 
.2994 
.3026 

3.4874 
3.4495 
3.4124 
3.3759 
33402 
3.3052 

.9613 
.9605 
.9596 
.9588 
.9580 
.9572 

00  74 

50 
40 
30 
20 
10 

17  00 

10 
20 
30 
40 
50 

.2924 
.2952 
.2979 
.3007 
.3035 
.3062 

.3057 
.3089 
.3121 
.3153 
.3185 
.3217 

3.2709 
3.2371 
3.2041 
3.1716 
3.1397 
3.1084 

.9563 
.9555 
.9546 
.9537 
.9528 
.9520 

00  73 

50 
40 
30 
20 
10 

18  00 

.3090 

.3249 

3.0777 

.9511 

00  72 

N.  Cos. 

N.  Cot. 

S.  Tan. 

N.  Sin. 

r   ° 

[93] 


°   f 

N.  Sin. 

N.  Tan. 

N.  Cot. 

N.  Cos. 

18  00 
10 
20 
30 
40 
50 

.3090 
.3118 
.3145 
.3173 
.3201 
.3228 

.3249 
.3281 
.3314 
.3346 
.3378 
.3411 

3.0777 
3.0475 
3.0178 
2.9887 
2.9600 
2.9319 

.9511 
.9502 
.9492 
.9483 
.9474 
.9465 

00  72 
50 
40 
30 
20 
10 

19  00 
10 
20 
30 

40 
^  50 

.3256 
.3283 
.3311 
.3338 
.3365 
.3393 

.3443 
.3476 
.3508 
.3541 
.3574 
.3607 

2.9042 
2.8770 
2.8502 
2.8239 
2.7980 
2.7725 

.9455 
.9446 
.9436 
.9426 
.9417 
.9407 

00  71 

50 
40 
30 
20 
10 

20  00 

10 
20 
30 
40 
50 

.3420 
.3448 
"  .3475 
.3502 
.3529 
.3557 

.3640 
.3673 
.3706 
.3739 
.3772 
.3805 

2.7475 
2.7228 
2.6985 
2.6746 
2.6511 
2.6279 

.9397 
.9387 
.9377 
.9367 
.9356 
.9346 

00  70 
50 
40 
30 
20 
10 

21  00 

10 
20 
30 
40 
50 

.3584 
.3611 
.3638 
.3665 
.3692 
.3719 

.3839 
.3872 
.3906 
.3939 
.3973 
.4006 

2.6051 
2.5826 
2.5605 
2.5386 
2.5172 
2.4960 

.9336 
.9325 
.9315 
.9304 
.9293 
.9283 

00  69 

50 
40 
30 
20 
10 

22  00 

10 
20 
30 
40 
50 

.3746 
.3773 
.3800 
.3827 
.3854 
.3881 

.4040 
.4074 
.4108 
.4142 
.4176 
.4210 

2.4751 
2.4545 
2.4342 
2.4142 
2.3945 
2.3750 

.9272 
.9261 
.9250 
.9239  . 
.9228 
.9216 

00  68 

50 
40 
30 
20 
10 

23  00 

10 
20 
30 

40 
50 

.3907 
.3934 
.3961 
.3987 
.4014 
.4041 

.4245 
.4279 
.4314 
.4348 
.4383 
.4417 

2.3559 
2.3369 
2.3183 
2.2998 
2.2817 
2.2637 

.9205 
.9194 
.9182 
.9171 
.9159 
.9147 

00  67 

50 
40 
30 
20 
10 

24  00 
10 
20 
30 
40 
50 

.4067 
.4094 
.4120 
.4147 
.4173 
.4200 

.4452 
.4487 
.4522 
.4557 
.4592 
.4628 

2.2460 
2.2286 
2.2113 
2.1943 
2.1775 
2.1609 

.9135 
.9124 
.9112 
.9100 
.9088 
.9075 

00  66 
50 
40 
30 
20 
10 

25  00 

10 
20 
30 
40 
50 

.4226 
.4253 
.4279 
.4305 
.4331 
.4358 

.4663 
.4699 
.4734 
4770 
.4806 
.4841 

2.1445 
2.1283 
2.1123 
2.0965 
2.0809 
2.0655 

.9063 
.9051 
.9038 
.9026 
.9013 
.9001  ' 

00  65 

50 
40 
30 
20' 
10 

26  00 

10 
20 
30 

40 
50 

.4384 
.4410 
.4436 
.4462 
.4488 
.4514 

.4877 
.4913 
.4950 
.4986 
.5022 
.5059 

2.0503 
2.0353 
2.0204 
2.0057 
1.9912 
1.9768 

.8988 
.8975 
.8962 
.8949 
.8936 
.8923 

00  64 

50 
40 
30 
20 
10 

27  00 

.4540 

.5095 

1.9626 

.8910 

00  63 

5.  Cos. 

X.  Cot. 

N.  Tan. 

N.  Sin. 

r   o 

[94] 


o    / 

N.  Sin. 

N.  Tan. 

N.  Cot. 

N.  Cos. 

27  00 

10 
20 
30 
40 
50 

.4540 
.4566 
.4592 
.4617 
.4643 
.4669 

.5095 
.5132 
.5169 
.5206 
.5243 
.5280 

'  1.9626 
1.9486 
1.9347 
1.9210 
1.9074 
1.8940 

.8910 
.8897 
.8884 
.8870 
.8857 
.8843 

00  63 

50 
40 
30 
20 
10 

28  00 

10 
20 
30 
40 
50 

.4695 
.4720 
.4746 
.4772 
.4797 
.4823 

.5317 
.5354 
.5392 
.5430 
.5467 
.5505 

1.8807 
1.8676 
1.8546 
1.8418 
1.8291 
1.8165 

.8829 
.8816 
.8802 
.8788 
.8774 
.8760 

00  62 

50 
40 
30 
20 
10   • 

29  00 

10 
20 
30 
40 
50 

.4848 
.4874 
.4899 
.4924 
.4950 
.4975 

.5543 
.5581 
.5619 
.5658 
.5696 
.5735 

1.8040 
1.7917 
1.7796 
1.7675 
1.7556 
1.7437 

.8746 
.8732 
.8718 
.8704 
.8689 
.8675 

00  61 

50 
40 
30 
20 
10 

30  00 

10 
20 
30 
40 
50 

.5000 
.5025 
.5050 
.5075 
.5100 
.5125 

.5774 
.5812 
.5851 
.5890 
.5930 
.5969 

1.7321 
1.7205 
1.7090 
1.6977 
1.6864 
1.6753 

.8660  .  , 
.8646 
.8631 
.8616 
.8601 
.8587 

00  60 

50 
40 
30 
20 
10 

31  00 

10 
20 
30 
40 
50 

.5150 
.5175 
.5200 
.5225 
.5250 
.5275 

.6009 
.6048 
.6088 
.6128 
.6168 
.6208 

1.6643 
1.6534 
1.6426 
1.6319 
.1.6212 
1.6107 

.8572 
.8557 
.8542 
.8526 
.8511 
.8496 

00  59 

50 
40 
30 
20 
10 

32  00 

10 
20 
30 
40 
50 

.5299 

.5324 
.5348 
.5373 
.5398 
.5422 

.6249 
.6289 
.6330 
.6371 
.6412 
.6453 

1.6003 
1.5900 
1.5798 
1.5697 
1.5597 
1.5497 

.8480 
.8465 
.8450 
.8434 
.8418 
.8403 

00  58 

50 
40  " 
30 
20 
10 

33  00 

10 
20 
30 
40 
50 

.5446 
•  .5471 
.5495 
.5519  ' 
.5544 
.5568 

.6494 
.6536 
.6577 
.6619 
.6661 
.6703 

1.5399 
1.5301 
1.5204 
1.5108 
1.5013 
1.4919 

.8387 
.8371 
.8355 
.8339 
.8323 
.8307 

00  57 

50 
40 
30 
20 
10 

34  00 

10 
20 
30 
40 
50 

.5592 
.5616 
.5640 
.5664 
.5688 
.5712 

.6745 
.6787 
.6830 
.6873 
.6916 
.6959 

1.4826 
1.4733 
1.4641 
1.4550 
1.4460 
1.4370 

.8290 
.8274 
.8258 
.8241 
.8225 
.8208 

00  56 

50 
40 
30 
20 
10 

35  00 

10 
20 
30 
40 
50 

.5736 
.5760 
.'5783 
.5807 
.5831 
.5854 

.7002 
.7046 
.7089 
.7133 
.7177 
.7221 

14281 
1.4193 
1.4106  . 
1.4019 
1.3934 
1.3848 

.8192 
.8175 
.8158 
.8141 
.8124 
.8107 

00  55 

50 
40 
30 
20 
10 

36  00 

.5878 

.7265 

1.3764 

.8090 

00  54 

N.  Cos. 

N.  Cot. 

N.  Tan. 

N.  Sin. 

t    0 

[95] 


o   f 

N.  Sin. 

N.  Tan. 

N.  Cot. 

N.  Co& 

36  00 

10 
20 
30 
40 
50 

.5878 
.5901 
.5925 
.5948 
.5972 
.5995 

.7265 
.7310 
.7355 
.7400 
.7445 
.7490 

1.3764 
1.3680 
1.3597 
1.3514 
1.3432 
1.3351 

.8090 
.8073 
.8056 
.8039 
.8021 
.8004 

00  54 
50 
40 
30 
20 
10 

37  00 

10 
20 
30 
40 
50 

.6018 
.6041 
.6065 
.6088 
.6111 
.6134 

.7536 
.7581 
.7627 
.7673 
.7720 
.7766 

1.3270 
1.3190 
1.3111 
1.3032 
1.2954 
1.2876 

.7986 
.7969^ 
.7951 
.7934 
.7916 
.7898 

00  53 

50 
40 
30 
20 
10 

38  00 

10 
20 
30 
40 
50 

.6157 
.6180 
.6202 
.6225 
.6248 
.6271 

.7813 
.7860 
.7907 
.7954 
.8002 
.8050 

1.2799 
1.2723 
1.2647 
1.2572 
1.2497 
1.2423 

.7880 
.7862 
.7844 
.7826 
.7808 
.7790 

00  52 

50 
40 
30 
20 
10 

39  00 

10 
20 
30 
40 
50 

.6293 
.6316 
.6338 
.6361 
.6383 
.6406 

.8098 
.8146 
.8195 
.8243 
.8292 
.8342 

1.2349 
1.2276 
1.2203 
1.2131 
1.2059 
1.1988 

.7771 
.7753 
.7735 
.7716 
.7698 
.7679 

00  51 

50 
40 
30 
20 
10 

40  00 

10 
20 
30 
40 
50 

.6428 
.6450 
.6472 
.6494 
.6517 
.6539 

.8391 
.8441 
.8491 
.8541 
.8591 
.8642 

1.1918 
1.1847 
1.1778 
1.1708 
1.1640 
1.1571 

.7660 
.7642 
.7623 
.7604 
.7585 
.7566 

00  50 

50 
40 
30 
20 
10 

41  00 

10 
20 
30 
40 
50 

.6561 
.6583 
.6604 
.6626 
.6648 
.b670 

.8693 
.8744 
.8796 
.8847 
.8899 
.8952 

1.1504 
1.1436 
1.1369 
'  1.1303 
1.1237 
1.1171 

.7547 
.7528 
.7509 
.7490 
.7470 
.7451 

00  49 

50 
40 
30 
20 
10 

42  00 
10 
20 
30 

40 
50 

.6691 
.6713 
.6734 
.6756 
.6777 
.6799 

.9004 
.9057 
.9110 
.9163 
.9217 
.9271 

1.110.6 
1.1041 
1.0977 
1.0913 
1.0850 
1.0786 

?431 
.7412 
.7392 
.7373 
.7353 
.7333 

00'  48 
50 
40 
30 
20 
10 

43  00 

10 
20 
30 

40 
50 

.6820 
.6841 
.6862 
.6884 
.6905 
.6926 

.9325 
.9380 
.9435 
.9490 
.9545 
.9601 

1.0724 
1.0661 
1,0599 
1.0538 
1.0477 
1.0416 

.7314 
.7294 
.7274 
.7254 
.7234 
.7214 

00  47 
50 
40 
30 
20 
10 

44  00 
10 

20 
30 
40 
50 

.6947 
.6967 
.6988 
.7009 
.7030 
.7050 

.9657 
.9713 
.9770 
.9827 
.9884 
.9942 

1.0355 
1.0295 
1.0235 
1.0176 
1.0117 
1.0058 

.7193 
.7173 
.7153 
.7133 
.7112 
.7092 

00  46 
50 
40 
30 
20 
10 

45  00 

.7071 

1.0000 

1.0000 

.7071 

00  45 

N.  Cos. 

ir.  Cot. 

N.  Tan. 

N.  Sin. 

/   o 

[96] 


TABLE  VI 


FOUR-PLACE  LOGARITHMS 


OF 


NUMBERS   1-2000 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

1 
2 
3 

0000 

0000 

3010 

4771 

6021 

6990 

7782 

8451 

9031 

9542 

0000 
3010 
4771 

0414 
3222 
4914 

0792 
3424 
5051 

1139 
3617 
5185 

1461 
3802 
5315 

1761 
3979 
5441 

2041 
4150 
5563 

2304 
4314 
5682 

2553 
4472 
5798 

2788 
4624 
5911 

4 
5 
6 

6021 
6990 
7782 

6128 
7076 
7853 

6232 
7160 
7924 

6335 
7243 
7993 

6435 
7324 
8062 

6532 
7404 
8129 

6628 
7482 
8195 

6721 
7559 
8261 

6812 
7634 
8325 

6902 
7709 
8388 

7 
8 
9 
10 

11 
12 
13 

8451 
9031 
9542 

8513 
9085 
9590 

8573 
9138 
9638 

8633 
9191 
9685 

8692 
9243 
9731 

8751 
9294 
9777 

8808 
9345 
9823 

8865 
9395 
9868 

8921 
9445 
9912 

8976 
9494 
9956 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

0414 
0792 
1139 

0453 
0828 
1173 

0492 
0864 
1206 

0531 
0899 
1239 

0569 
0934 
1271 

0607 
0969 
1303 

0645 
1004 
1335 

0682 
1038 
1367 

0719 
1072 
1399 

0755 
1106 
1430 

14 
15 
16 

1461 
1761 
2041 

1492 
1790 
2068 

1523 
1818 
2095 

1553 
1847 
2122 

1584 
1875 
2148 

1614 
1903 
2175 

1644 
1931 
2201 

1673 
1959 
2227 

1703 
1987 
2253 

1732 
2014 
2279 

17 
18 
19 
20 

21 
22 
23 

2304 
2553 
2788 

2330 
2577 
2810 

2355 
2601 
2833 

2380 
2625 
2856 

2405 
2648 
2878 

2430 
2672 
2900 

2455 
2695 
2923 

2480 
2718 
2945 

2504 
2742 
2967 

2529 
2765 
2989 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

3222 
3424 
3617 

3243 
3444 
3636 

3263 
3464 
3655 

3284 
3483 
3674 

3304 
3502 
3692 

3324 
3522 
3711 

3345 
3541 
3729 

3365 
3560 
3747 

3385 
3579 
3766 

3404 
3598 
3784 

24 
25 
26 

3802 
3979 
4150 

3820 
3997 
4166 

3838 
4014 
4183 

3856 
4031 
4200 

3874 
4048 
4216 

3892 
4065 
4232 

3909 
4082 
4249 

3927 
4099 
4265 

3945 
4116 
4281 

3962 
4133 
4298 

27 
28 
29 
30 
31 
32 
33 

4314 
4472 
4624 

4330 
4487 
4639 

4346 
4502 
4654 

4362 
4518 
4669 

4378 
4533 
4683 

4393 
4548 
4698 

4409 
4564 
4713 

4425 
4579 
4728 

4440 
4594 
4742 

4456 
4609 
4757 

4771 

4786 

4800  1  4814 

4829 

4843 

4857 

4871 

4886 

4900 

4914 
5051 
5185 

4928 
5065 
5198 

4942   4955 
5079  1  5092 
5211   5224 

4969 
5105 
5237' 

4983 
5119 
5250 

4997 
5132 
5263 

5011 
5145 
5276 

5024 
5159 
5289 

5038 
5172 
5302 

34 
35 
36 

5315 
5441 
5563 

5328 
5453 
5575 

5340 
5465 
5587 

5353 
5478 
5599 

5366 
5490 
5611 

5378 
5502 
5623 

5391 
5514 
5635 

5403 
5527 
5647 

5416 
5539 
5658 

5428 
5551 
5670 

37 
38 
39 
40 

41 
42 
43 

5682 
5798 
5911 

5694 
5809 
5922 

5705. 
582  ll 
5933^ 

L5717 
K832 
~5944 

5729 
5843 
5955 

5740 
5855 
5966 

5752 
5866 
5977 

5763 
5877 
5988 

5775 
5888 
5999 

5786 
5900 
6010 

6021 

6031 

6042  |  6053 

6064 

6075 

6085 

6096 

6107 

6117 

6128 
6232 
6335 

6138 
6243 
6345 

6149 
6253 
6355 

6160 
6263 
6365 

6170 
6274 
6375 

6180 
6284 
6385 

6191 
6294 
6395 

6201 
6304 
6405 

6212 
6314 
6415 

6222 
6325 
6425 

44 
45 
46 

6435 
6532 
6628 

6444 
6542 
6637 

6454 
6551 
6646 

6464 
6561 
6656 

6474 
6571 
6665 

6484 
6580 
6675 

6493 
6590 
6684 

6503 
6599 
6693 

6513 
6609 
6702 

6522 
6618 
6712 

47 
48 
49 
50 

6721 
6812 
6902 

6730 
6821 
6911 

6739 
6830 
6920 

6749 
6839 
6928 

6758 
6848 
6937 

6767 
6857 
6946 

6776 
6866 
6955 

6785 
6875 
6964 

6794 
6884 
6972 

6803 
6893 
6981 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

[98] 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973, 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293  . 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8803 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

90G9 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9513 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9866 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9323 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

'9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

100 

0000 

0004 

0009 

0013 

0017 

0022 

0026 

0030 

0035 

0039 

N. 

O 

1 

2 

3   |   4 

5 

6 

7 

8 

9 

[99] 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

0000 

0004 

0009 

0013 

0017 

0022 

0026 

0030 

0035 

0039 

101 

0043 

0048 

0052 

0056 

0060 

0065 

0069 

0073 

0077 

0082 

102 

0086 

0090 

0095 

0099 

0103 

0107 

0111 

0116 

0120 

0124 

103 

0128 

0133 

0137 

0141 

0145 

0149 

0154 

0158 

0162 

0166 

104 

0170 

0175 

0179 

0183 

0187 

0191 

0195 

0199 

0204 

0208 

105 

0212 

0216 

0220 

0224 

0228 

0233 

0237 

0241 

0245 

0249 

106 

0253 

0257 

0261 

0265 

0269 

0273 

0278 

0282 

0286 

0290 

107 

0294 

0298 

0302 

0306 

0310 

0314 

0318 

0322 

0326 

0330 

108 

0334 

0338 

0342 

0346 

0350 

0354 

0358 

0362 

0366 

0370 

109 

0374 

0378 

0382 

0386 

0390 

0394 

0398 

0402 

0406 

0410 

110 

0414 

0418 

0422 

0426 

0430 

0434 

0438 

0441 

0445 

0449 

111 

0453 

0457 

0461 

0465 

0469 

0473 

0477 

0481 

0484 

0488 

112 

0492 

0496 

0500 

0504 

0508 

0512 

0515 

0519 

0523 

0527 

113 

0531 

0535 

0538 

0542 

0546 

0550 

0554 

0558 

0561 

0565 

114 

0569 

0573 

0577 

0580 

0584 

0588 

0592 

0596 

0599 

0603 

115 

0607 

0611 

0615 

0618 

0622 

0626 

0630 

0633 

0637 

0641 

116 

0645 

0648 

0652 

0656 

0660 

0663 

0667 

0671 

0674 

0678 

117 

0682 

0686 

0689 

0693 

0697 

0700 

0704 

0708 

0711 

0715 

118 

0719 

0722 

0726 

0730 

0734 

0737 

0741 

0745 

0748 

0752 

119 

0755 

0759 

0763 

0766 

0770 

0774 

0777 

0781 

0785 

0788 

120 

0792 

0795 

0799 

0803 

0806 

0810 

0813 

0817 

0821 

0824 

121 

0828 

0831 

0835 

0839 

0842 

0846 

0849 

0853 

0856 

0860 

122 

0864 

0867 

0871 

0874 

0878 

0881 

0885 

0888 

0892 

0896 

123 

0899 

0903 

0906 

0910 

0913 

0917 

0920 

0924 

0927 

0931 

124 

0934 

0938 

0941 

0945 

0948 

0952 

0955 

0959 

0962 

0966 

125 

0969 

0973 

0976 

0980 

0983 

0986 

0990 

0993 

0997 

1000 

126 

1004 

1007 

1011 

1014 

1017 

1021 

1024 

1028 

1031 

1035 

127 

1038 

1041 

1045 

1048 

1052 

1055 

1059 

1062 

1065 

1069 

128 

1072 

1075 

1079 

1082 

1086 

1089 

1092 

1096 

1099 

1103 

129 

1106 

1109 

1113 

1116 

1119 

1123 

1126 

1129 

1133 

1136 

130 

1139 

1143 

1146 

1149 

1153 

1156 

1159 

1163 

1166 

1169 

131 

1173 

1176 

1179 

1183 

1186 

4189 

1193 

1196 

1199 

1202 

132 

1206 

1209 

1212 

1216 

1219 

1222 

1225 

1229 

1232 

1235 

133 

1239 

1242 

1245 

1248 

1252 

1255 

1258 

1261 

1265 

1268 

134 

1271 

1274 

1278 

1281 

1284 

1287 

1290 

1294 

1297 

1300 

135 

1303 

1307 

1310 

1313 

1316 

1319 

1323 

1326 

1329 

1332 

136 

1335 

1339 

1342 

1345 

1348 

1351 

1355 

1358 

1361 

1364 

137 

1367 

1370 

1374 

1377 

1380 

1383 

1386 

1389 

1392 

1396 

138 

1399 

1402 

1405 

1408 

1411 

1414 

1418 

1421 

1424 

1427 

139 

1430 

1433 

1436 

1440 

1443 

1446 

1449 

1452 

1455 

1458 

140 

1461 

1464 

1467 

1471 

1474 

1477 

1480 

1483 

1486 

1489 

141 

1492 

1495 

1498 

1501 

1504 

1508 

1511 

1514 

1517 

1520 

142 

1523 

1526 

1529 

1532 

1535 

1538 

1541 

1544 

1547 

1550 

143 

1553 

1556 

1559 

1562 

1565 

1569 

1572 

1575 

1578 

1581 

144 

1584 

1587 

1590 

1593 

1596 

1599 

1602 

1605 

1608 

1611 

145 

1614 

1617 

1620 

1623 

1626 

1629 

1632 

1635 

1638 

1641 

146 

1644 

1647 

1649 

1652 

1655 

1658 

1661 

1664 

1667 

1670 

147 

1673 

1676 

1679 

1682 

1685 

1688 

1691 

1694 

1697 

1700 

148 

1703 

1706 

1708 

1711 

1714 

1717 

1720 

1723 

1726 

1729 

149 

1732 

1735 

1738 

1741 

1744 

1746 

1749 

1752 

1755 

1758 

150 

1761 

1764 

1767 

1770 

1772 

1775 

1778 

1781 

1784 

1787 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

[100] 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

150 

1761 

1764 

1767 

1770 

1772 

1775 

1778 

1781 

1784 

1787 

151 

1790 

1793 

1796 

1798 

1801 

1804 

1807 

1810 

1813 

1816 

152 

1818 

,  1821 

1824 

1827 

1830 

1833 

1836 

1838 

1841 

1844 

153 

1847 

1850 

1853 

1855 

1858 

1861 

1864 

1867 

1870 

1872 

154 

1875 

1878 

1881 

1884 

1886 

1889 

1892 

1895 

1898 

1901 

155 

1903 

1906 

1909 

1912 

1915 

1917 

1920 

1923 

1926 

1928 

156 

1931 

1934 

1937 

1940 

1942 

1945 

1948 

1951 

1953 

1956 

157 

1959 

1962 

1965 

1967 

1970 

1973 

1976 

1978 

1981 

1984 

158 

1987 

1989 

1992 

1995 

1998 

2000 

2003 

2006 

2009 

2011 

159 

2014 

2017 

2019 

2022 

2025 

2028 

2030 

2033 

2036 

2038 

160 

2041 

2044 

2047 

2049 

2052 

2055 

2057 

2060 

2063 

2066 

161 

2068 

2071 

2074 

2076 

2079 

2082 

2084 

2087 

2090 

2092 

162 

2095 

2098 

2101 

2103 

2106 

2109 

2111 

2114 

2117 

2119 

163 

2122 

2125 

2127 

2130 

2133 

2135 

2138 

2140 

2143 

2146 

164 

2148 

2151 

2154 

2156 

2159 

2162 

2164 

2167 

2170 

2172 

165 

2175 

2177 

2180 

2183 

2185 

2188 

2191 

2193 

2196 

2198 

166 

2201 

2204 

2206 

2209 

2212 

2214 

2217 

2219 

2222 

2225 

167 

2227 

2230 

2232 

2235 

2238 

2240 

2243 

2245 

2248 

2251 

168 

2253 

2256 

2258 

2261 

2263 

2266 

2269 

2271 

2274 

2276 

169 

2279 

2281 

2284 

2287 

2289 

2292 

2294 

2297 

2299 

2302 

170 

2304 

2307 

2310 

2312 

2315 

2317 

2320 

2322 

2325 

2327 

171 

2330 

2333 

2335 

2338 

2340 

2343 

2345 

2348 

2350 

2353 

172 

2355 

2358 

2360 

2363 

2365 

2368 

2370 

2373 

2375 

2378 

173 

2380 

2383 

2385 

2388 

2390 

2393 

2395 

2398 

2400 

2403 

174 

2405 

2408 

2410 

2413 

2415 

2418 

2420 

2423 

2425 

2428 

175 

2430 

2433 

2435 

2438 

2440 

2443 

2445 

2448 

2450 

2453 

176 

2455 

2458 

2460 

2463 

2465 

2467 

2470 

2472 

2475 

2477 

177 

2480 

2482 

2485 

2487 

2490 

2492 

2494 

2497 

2499 

2502 

178 

2504 

2507 

2509 

2512 

2514 

2516 

2519 

2521 

2524 

2526 

179 

2529 

2531 

2533 

2536 

2538 

2541 

2543 

2545 

2548 

2550 

180 

2553 

2555 

2558 

2560 

2562 

2565 

2567 

2570 

2572 

2574 

181 

2577 

2579 

2582 

2584 

2586 

2589 

2591 

2594 

2596 

2598 

•182 

2601 

2603 

2605 

2608 

2610 

2613 

2615 

2617 

2620 

2622 

183 

2625 

2627 

2629 

2632 

2634 

2636 

2639 

2641 

2643 

2646 

184 

2648 

2651 

2653 

2655 

2658 

2660 

2662 

2665 

2667 

2669 

185 

2672 

2674 

2676 

2679 

2681 

2683 

2686 

2688 

2690 

2693 

186 

2695 

2697 

2700 

2702 

2704 

2707 

2709 

2711 

2714 

2716 

187 

2718 

2721 

2723 

2725 

2728 

2730 

2732 

2735 

2737 

2739 

188 

2742 

2744 

2746 

2749 

2751 

2753 

2755 

2758 

2760 

2762 

189 

2765 

.2767 

2769 

2772 

2774 

2776 

2778 

2781 

2783 

2785 

190 

2788 

2790 

2792 

2794 

2797 

2799 

2801 

2804 

2806 

2808 

191 

2810 

2813 

2815 

2817 

2819 

2822 

2824 

2826 

2828 

2831 

192 

2833 

2835 

2838 

2840 

2842 

2844 

2847 

2849 

2851 

2853 

193 

2856 

2858 

2860 

2862 

2865 

2867 

2869 

2871 

2874 

2876 

194 

2878 

2880 

2883 

2885 

2887 

2889 

2891 

2894 

2896 

2898 

195 

2900 

2903 

2905 

2907 

2909 

2911 

2914 

2916 

2918 

2920 

196 

2923 

2925 

2927 

2929 

2931 

2934 

2936 

2938 

2940 

2942 

197 

2945 

2947 

2949 

2951 

2953 

2956 

2958 

2960 

2962 

2964 

198 

2967 

2969 

2971 

2973 

2975 

2978 

2980 

2982 

2984 

2986 

199 

2981? 

2991 

2993 

2995 

2997 

2999 

3002 

3004 

3006 

3008 

200 

3010 

3012 

3015 

3017 

3019 

3021 

3023 

3025 

3028 

3030 

N.  1 

.—  —  _ 
0 

— 

«I^B««™M™ 

2 

3 

4 

—  — 

—  —  m 

[101] 


TABLE  VII 


FOUR-PLACE  LOGARITHMS 


OP   THE 


TRIGONOMETRIC  FUNCTIONS 


FOR    THE 


DECIMALLY  DIVIDED   DEGREE 


[103] 


L.  Sin. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

••—  —^— 

o°.o 

H^HKXMB 

—  00 

6.2419 

5429 

7190 

8439 

9408 

*0200 

*0870 

*1450 

*1961 

*2419 

89.9 

0.1 

7.2419 

2833 

3211 

3558 

3880 

4180 

4460 

4723 

4971 

5206 

5429 

89.8 

0.2 

7.5429 

5641 

5843 

6036 

6221 

6398 

6568 

6732 

6890 

7043 

7190 

89.7 

0.3 

7.7190 

7332 

7470 

7604 

7734 

7859 

7982 

8101 

8217 

8329 

8439 

89.6 

0.4 

7.8439 

8547 

8651 

8753 

8853 

8951 

9046 

9140 

9231 

9321 

9408 

89.5 

0.5 

7.9408 

9494 

9579 

9661 

9743 

9822 

9901 

9977 

*0053 

*0127 

*0200 

89.4 

0.6 

8.0200 

0272 

0343 

0412 

0480 

0548 

0614 

0679 

0744 

0807 

0870 

89.3 

0.7 

8.0870 

0931 

0992 

1052 

1111 

1169 

1227 

1284 

1340 

1395 

1450 

89.2 

0.8 

8.1450 

1503 

1557 

1609 

1661 

1713 

1764 

1814 

1863 

1912 

1961 

89.1 

0.9 

8.1961 

2009 

2056 

2103 

2150 

2196 

2241 

2286 

2331 

2375 

2419 

89°.0 

1°.0 

8.2419 

2462 

2505 

2547 

2589 

2630 

2672 

2712 

2753 

2793 

2832 

88.9 

1.1 

8.2832 

2872 

2911 

2949 

2988 

3025 

3063 

3100 

3137 

3174 

3210 

88.8 

1.2 

8.3210 

3246 

3282 

3317 

3353 

3388 

3422 

3456 

3491 

3524 

3558 

88.7 

1.3 

8.3558 

3591 

3624 

3657 

3689 

3722 

3754 

3786 

3817 

3848 

3880 

88.6 

1.4 

8.3880 

3911 

3941 

3972 

4002 

4032 

4062 

4091 

4121 

4150 

4179 

88.5 

1.5 

8.4179 

4208 

4237 

4265 

4293 

4322 

4349 

4377 

4405 

4432 

4459 

88.4 

1.6 

8.4459 

4486 

4513 

4540 

4567 

4593 

4619 

4645 

4671 

4697 

4723 

88.3 

1.7 

8.4723 

4748 

4773 

4799 

4824 

4848 

4873 

4898 

4922 

4947 

4971 

88.2 

1.8 

8.4971 

4995 

5019 

5043 

5066 

5090 

5113 

5136 

5160 

5183 

5206 

88.1 

1.9 

8.5206 

5228 

5251 

5274 

5296 

5318 

5340 

5363 

5385 

5406 

5428 

88°.0 

2°.0 

8.5.428 

5450 

5471 

5493 

5514 

5535 

5557 

5578 

5598 

5619 

5640 

87.9 

2.1 

8.5640 

5661 

5681 

5702 

5722 

5742 

5762 

5782 

5802 

5822 

5842 

87.8 

2.2 

8.5842 

5862 

5881 

5901 

5920 

5939 

5959 

5978 

5997 

6016 

6035 

87.7 

2.3 

8.6035 

6054 

6072 

6091 

6110 

6128 

6147 

6165 

6183 

6201 

6220 

87.6 

2.4 

8.6220 

6238 

6256 

6274 

6291 

6309 

6327 

6344 

6362 

6379 

6397 

87.5 

2.5 

8.6397 

6414 

6431 

6449 

6466 

6483 

6500 

6517 

6534 

6550 

6567 

87.4 

2.6 

8.6567 

6584 

6600 

6617 

6633 

6650 

6666 

6682 

6699 

6715 

6731 

87.3 

2.7 

8.6731 

•6747 

6763 

6779 

6795 

6810 

6826 

6842 

6858 

6873 

6889 

87.2 

2.8 

8.6889 

6904 

6920 

6935 

6950 

6965 

6981 

6996 

7011 

7026 

7041 

87.1 

2.9 

8.7041 

7056 

7071 

7086 

7100 

7115 

7130 

7144 

7159 

7174 

7188 

87°.0 

3°.0 

8.7188 

7202 

7217 

7231 

7245 

7260 

7274 

7288 

7302 

7316 

7330 

86.9 

3.1 

8.7330 

7344 

7358 

7372 

7386 

7400 

7413 

7427 

7441 

7454 

7468 

86.8 

3.2 

8.7468 

7482 

7495 

7508 

7522 

7535 

7549 

7562 

7575 

7588 

7602 

86.7 

3.3 

8.7602 

7615 

7628 

7641 

7654 

7667 

7680 

7693 

7705 

7718 

7731 

86.6 

3.4 

8.7731 

•7744 

7756 

7769 

7782 

7794 

7807 

7819 

7832 

7844 

7857 

86.5 

3.5 

8.7857 

7869 

7881 

7894 

7906 

7918 

7930 

7943 

7955 

7967 

7979 

86.4 

3.6 

8.7979 

7991 

8003 

8015 

8027 

8039 

8051 

8062 

8074 

8086 

8098 

86.3 

3.V 

8.8098 

8109 

8121 

8133 

8144 

8156 

8168 

8179 

8191 

8202 

8213 

86.2 

3.8 

8.8213 

8225 

8236 

8248 

8259 

8270 

8281 

8293 

8304 

8315 

8326 

86.1 

3.9 

8.8326 

8337 

8348 

8359 

8370 

8381 

8392 

8403 

8414 

8425 

8436 

86°.0 

4°.0 

8.8436 

8447 

8457 

8468 

8479 

8490 

8500 

8511 

8522 

8532 

8543 

85.9 

4.1 

8.8543 

8553 

8564 

8575 

8585 

8595 

8606 

8616 

8627 

8637 

8647 

85.8 

4.2 

8.8647 

8658 

8668 

8678 

8688 

8699 

8709 

8719 

8729 

8739 

8749 

85.7 

4.3 

8.8749 

8759 

8769 

8780 

8790 

8799 

8809 

8819 

8829 

8839 

8849 

85.6 

4.4 

8.8849 

8859 

8869 

8878 

8888 

8898 

8908 

8917 

8927 

8937 

8946 

85.5 

I 

4.5 

8.8946 

8956 

8966 

8975 

8985 

8994 

9004 

9013 

9023 

9032 

9042 

85.4 

4.6 

8.9042 

9051 

9060 

9070 

9079 

9089 

9098 

9107 

9116 

9126 

9135 

85.3 

4.7 

8.9135 

9144 

9153 

9162 

9172 

9181 

9190 

9199 

9208 

9217 

9226 

85.2 

4.8 

8.9226 

9235 

9244 

9253 

9262 

9271 

9280 

9289 

9298 

9307 

9315 

85.1 

4.9 

8.9315 

MHHIHB 

9324 

^••i^MM 

9333 

9342 

9351 

9359 

9368 

9377 

9386 

9394 

9403 

85°.O 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

L.  Cos. 

[104] 


L.  Sin. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

5°.0 

8.9403 

9412 

9420 

9429 

9437 

9446 

9455 

9463 

9472 

9480 

9489 

84.9 

5.1 

8.9489 

9497 

9506 

9514 

9523 

9531 

9539 

9548 

9556 

9565 

9573 

84.8 

5.2 

8.9573 

9581 

9589 

9598 

9606 

9614 

9623 

9631 

9639 

9647 

9655 

84.7 

5.3 

8.9655 

9664 

9672 

9680 

9688 

9696 

9704 

9712 

9720 

9728 

9736 

84.6 

5.4 

8.9736 

9744 

9752 

9760 

9768 

9776 

9784 

9792 

9800 

9808 

9816 

84.5 

5.5 

8.9816 

9824 

9831 

9839 

9847 

9855 

9863 

9870 

9878 

9886 

9894 

84.4 

5.6* 

8.9894 

9901 

9909 

9917 

9925 

9932 

9940 

9948 

9955 

9933 

9970 

84.3 

5.7 

8.9970 

9978 

9986 

9993 

*0001 

*0008 

*0016 

*0023 

*0031 

*0038 

*0046 

84.2 

5.8 

9.0046 

0053 

0061 

0068 

0075 

0083 

0090 

0098 

0105 

0112 

0120 

84.1 

5.9 

9.0120 

0127 

0134 

0142 

0149 

0156 

0163 

0171 

0178 

0185 

0192 

84°.O 

6°.0 

9.0192 

0200 

0207 

0214 

0221 

0228 

0235 

0243 

0250 

0257 

0264 

83.9 

6.1 

9.0264 

0271 

0278 

0285 

0292 

0299 

0306 

0313 

0320 

0327 

0334 

83.8 

6.2 

9.0334 

0341 

0348 

0355 

0362 

0369 

0376 

0383 

0390 

0397 

0403 

83.7 

6.3 

9.0403 

0410 

0417 

0424 

0431 

0438 

0444 

0451 

0458 

0465 

0472 

83.6 

6.4 

9.0472 

0478 

0485 

0492 

0498 

0505 

0512 

0519 

0525 

0532 

0539 

83.5 

6.5 

9.0539 

0545 

0552 

0558 

0565 

0572 

0578 

0585 

0591 

0598 

0605 

83.4 

6.6 

9.0605 

0311 

0618 

0624 

0631 

0637 

0644 

0650 

0657 

0663 

0670 

83.3 

6.7 

9.0670 

0676 

0683 

0689 

0695 

0702 

0708 

0715 

0721 

0727 

0734 

83.2 

6.8 

9.0734 

0740 

0746 

0753 

0759 

0765 

0772 

0778 

0784 

0790 

0797 

83.1 

6.9 

9.0797 

0803 

0809 

0816 

0822 

0828 

0834 

0840 

0847 

0853 

0859 

83°.0 

7°.0 

9.0859 

0865 

0871 

0877 

0884 

0890 

0896 

0902 

0908 

0914 

0920 

82.9 

7.1 

9.0920 

0926 

0932 

0938 

0945 

0951 

0957 

0933 

0969 

0975 

0981 

82.8 

7.2 

9.0981 

0987 

0993 

0999 

1005 

1011 

1017 

1022 

1028 

1034 

1040 

82.7 

7.3 

9.1040 

1046 

1052 

1058 

1064 

1070 

1076 

1031 

1087 

1093 

1099 

82.6 

7.4 

9.1099 

1105 

1111 

1116 

1122 

1128 

1134 

1140 

1145 

1151 

1157 

82.5 

7.5 

9.1157 

1163 

1168 

1174 

1180 

1186 

1191 

"1197 

1203 

1208 

1214 

82.4 

7.6 

9.1214 

1220 

1226 

1231 

1237 

1242 

1248 

1254 

1259 

1265 

1271 

82.3 

7.7 

9.1271 

1276 

1282 

1287 

1293 

1299 

1304 

1310 

1315 

1321 

1326 

82.2 

7.8 

9.1326 

1332 

1337 

1343 

1348 

1354 

1359 

1365 

1370 

1376 

1381 

82.1 

7.9 

9.1381 

1387 

1392 

1398 

1403 

1409 

1414 

1419 

1425 

1430 

1436 

82°.0 

8°.0 

9.1436 

1441 

1446 

1452 

1457 

1462 

1468 

1473 

1478 

1484 

1489 

81.9 

8.1 

9.1489 

1494 

1500 

1505 

1510 

1516 

1521 

1526 

1532 

1537 

1542 

81.8 

8.2 

9.1542 

1547 

1553 

1558 

1563 

1568 

1574 

1579 

1584 

1589 

1594 

81.7 

8.3 

9.1594 

1600 

1605 

1610 

1615 

1620 

1625 

1631 

1636 

1641 

1646 

81.6 

8.4 

9.1646 

1651 

1656 

1661 

1666 

1672 

1677 

1682 

1687 

1692 

1697 

81.5 

8.5 

9.1697 

1702 

1707 

1712 

1717 

1722 

1727 

1732 

1737 

1742 

1747 

81.4 

8.6 

9.1747 

1752 

1757 

1762 

1767 

1772 

1777 

1782 

1787 

1792 

1797 

81.3 

8.7 

9.1797 

1802 

1807 

1812 

1817 

1822 

1827 

1832 

1837 

1842 

1847 

81.2 

8.8 

9.1847 

1851 

1856 

1861 

1866 

1871 

1876 

1881 

1886 

1890 

1895 

81.1 

8.9 

9.1895 

1900 

1905 

1910 

1915 

1919 

1924 

1929 

1934 

1939 

1943 

81°.O 

9°.0 

9.1943 

1948 

1953 

1958 

1962 

1967 

1972 

1977 

1981 

1986 

1991 

80.9 

9.1 

9.1991 

1996 

2000 

2005 

2010 

2015 

2019 

2024 

2029 

2033 

2038 

80.8 

9.2 

9.2038 

2043 

2047 

2052" 

2057 

2061 

2066 

2071 

2075 

2080 

2085 

80.7 

9.3 

9.2085 

2089 

2094 

2098 

2103 

2108 

2112 

2117 

2121 

2126 

2131 

80.6 

9.4 

9.2131 

2135 

2140 

2144 

2149 

2153 

2158 

2162 

2167 

2172 

2176 

80.5 

9.5 

9.2176 

2181 

2185 

2190 

2194 

2199 

2203 

2208 

2212 

2217 

2221 

80.4 

9.6 

9.2221 

2226 

2230 

2235 

2239 

2243 

2248 

2252 

2257 

2261 

2266 

80.3 

9.7 

9.2266 

2270 

2275 

2279 

2283 

2288 

2292 

2297 

2301 

2305 

2310 

80.2 

9.8 

9.2310 

2314 

2319 

2323 

2327 

2332 

2336 

2340 

2345 

2349 

2353 

80.1 

9.9 

9.2353 

••••••Ml 

2358 

2362 

2367 

2371 

2375 

2379 

2384 

2388 

2392 

2397 

8O°.0 

• 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

L.  Cos. 

[105] 


L.  Sin. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

•HHBm 

—  00 

—  — 
90° 

0° 

-00 

7.2419 

5429 

7190 

8439 

9408 

*0200 

*0870 

*1450 

*1961 

*2419 

89 

1 

8.2419 

2832 

3210 

3558 

3880 

4179 

4459 

4723 

4971 

5206 

5428 

88 

2 

8.5428 

5640 

5842 

6035 

6220 

6397 

6567 

6731 

6889 

7041 

7188 

87 

3 

8.7188 

7330 

7468 

7602 

7731 

7857 

7979 

8098 

8213 

8326 

8436 

86 

4 

8.8436 

8543 

8647 

8749 

8849 

8946 

9042 

9135 

9226 

9315 

9403 

85 

5 

8.9403 

9489 

9573 

9655 

9736 

9816 

9894 

9970 

*0046 

*0120 

*0192 

84 

6 

9.0192 

0264 

0334 

0403 

0472 

0539 

0605 

0670 

0734 

0797 

0859 

83 

7 

9.0859 

0920 

0981 

1040 

1099 

1157 

1214 

1271 

1326 

1381 

1436 

82 

8 

9.1436 

1489 

1542 

1594 

1646 

1697 

1747 

1797 

1847 

1895 

1943 

81 

9 

9.1943 

1991 

2038 

2085 

2131 

2176 

2221 

2266 

2310 

2353 

2397 

80° 

10° 

9.2397 

2439 

2482 

2524 

2565 

2606 

2647 

2687 

2727 

2767 

2806 

79 

11 

9.2806 

2845 

2883 

2921 

2959 

2997 

3034 

3070 

3107 

3143 

3179 

78 

12 

9.3179 

3214 

3250 

3284 

3319 

3353 

3387 

3421 

3455 

3488 

3521 

77 

13 

9.3521 

3554 

3586 

3618 

3650 

3682 

3713 

3745 

3775 

3806 

3837 

76 

14 

9.3837 

3867 

3897 

3927 

3957 

3986 

4015 

4044 

4073 

4102 

4130 

75 

15 

9.4130 

4158 

4186 

4214 

4242 

4269 

4296 

4323 

4350 

4377 

4403 

74 

16 

9.4403 

4430 

4456 

4482 

4508 

4533 

4559 

4584 

4609 

4634 

4659 

73 

17 

9.4659 

4684 

4709 

4733 

4757 

4781 

4805 

4829 

4853 

4876 

4900 

72 

18 

9.4900 

4923 

4946 

4969 

4992 

5015 

5037 

5060 

5082 

5104 

5126 

71 

19 

9.5126 

5148 

5170 

5192 

5213 

5235 

5256 

5278 

5299 

5320 

5341 

7O° 

20° 

9.5341 

5361 

5382 

5402 

5423 

5443 

5463 

5484 

5504 

5523 

5543 

69 

21 

9.5543 

5563 

5583 

5602 

5621 

5641 

5660 

5679 

5698 

5717 

5736 

68 

22 

9.5736 

5754 

5773 

5792 

5810 

5828 

5847 

5865 

5883 

5901 

5919 

67 

23 

9.5919 

5937 

5954 

5972 

5990 

6007 

6024 

6042 

6059 

6076 

6093 

66 

24 

9.6093 

6110 

6127 

6144 

6161 

6177 

6194 

6210 

6227 

6243 

6259 

65 

25 

9.6259 

6276 

6292 

6308 

6324 

6340 

6356 

6371 

6387 

6403 

6418 

64 

26 

9.6418 

6434 

6449 

6465 

6480 

6495 

6510 

6526 

6541 

6556 

6570 

63 

27 

9.6570 

6585 

6600 

6615 

6629 

6644 

6659 

6673 

6687 

6702 

6716 

62 

28 

9.6716 

6730 

6744 

6759 

6773 

6787 

6801 

6814 

6828 

6842 

6856 

61 

29 

9.6856 

6869 

6883 

6896 

6910 

6923 

6937 

6950 

6963 

6977 

6990 

60° 

30° 

9.6990 

7003 

7016 

7029 

7042 

7055 

7068 

7080 

7093 

7106 

7118 

59 

31 

9.7118 

7131 

7144 

7156 

7168 

7181 

7193 

7205 

7218 

7230 

7242 

58 

32 

9.7242 

7254 

7266 

7278 

7290 

7302 

7314 

7326 

7338 

7349 

7361 

57 

33 

9.7361 

7373 

7384 

7396 

7407 

7419 

7430 

7442 

7453 

7464 

7476 

56 

34 

9.7476 

7487 

7498 

7509 

7520 

7531 

7542 

7553 

7564 

7575 

7586 

55 

35 

9.7586 

7597 

7607 

7618 

7629 

7640 

7650 

7661 

7671 

7682 

7692 

54 

36 

9.7692 

7703 

7713 

7723 

7734 

7744 

7754 

7764 

7774 

7785 

7795 

53 

37 

9.7795 

7805 

7815 

7825 

7835 

7844 

7854 

7864 

7874 

7884 

7893 

52 

38 

9.7893 

7903 

7913 

7922 

7932 

7941 

7951 

7960 

7970 

7979 

7989 

51 

39 

9.7989 

7998 

8007 

8017 

8026 

8035 

8044 

8053 

8063 

8072 

8081 

50° 

40° 

9.8081 

8090 

8099 

8108 

8117 

8125 

8134 

8143 

8152 

8161 

8169 

49 

41 

9.8169 

8178 

8187 

8195 

8204 

8213 

8221 

8230 

8238 

8247 

8255 

48 

42 

9.8255 

8264 

8272 

8280 

8289 

8297 

8305 

8313 

8322 

8330 

8338 

47 

43 

9.8338 

8346 

8354 

8362 

8370 

8378 

8386 

8394 

8402 

8410 

8418 

46 

44 

9.8418 

8426 

8433 

8441 

8449 

8457 

8464 

8472 

8480 

8487 

8495 

45°- 

45° 

9.8495 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

L.  Cos. 

[106] 


L.  Sin. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

" 

9.8495 

45° 

45° 

9.8495 

8502 

8510 

8517 

8525 

8532 

8540 

8547 

8555 

8562 

8569 

44 

46 

9.8569 

8577 

8584 

8591 

8598 

8606 

8613 

8620 

8627 

8634 

8641 

43 

47 

9.8641 

8648 

8655 

8662 

8669 

8676 

8683 

8690 

8697 

8704 

8711 

42 

48 

9.8711 

8718 

8724 

8731 

8738 

8745 

8751 

8758 

8765 

8771 

8778 

41 

49 

9.8778 

8784 

8791 

8797 

8804 

8810 

8817 

8823 

8830 

8836 

8843 

40° 

50° 

9.8843 

8849 

8855 

8862 

8868 

8874 

8880 

8887 

8893 

8899 

8905 

39 

51 

9.8905 

8911 

8917 

8923 

8929 

8935 

8941 

8947 

8953 

8959 

8965 

38 

52 

9.8965 

8971 

8977 

8983 

8989 

8995 

9000 

9006 

9012 

9018 

9023 

37 

53 

9.9023 

9029 

9035 

9041 

9046 

9052 

9057 

9063 

9069 

9074 

9080 

36 

54 

9.9080 

9085 

9091 

9096 

9101 

9107 

9112 

9118 

9123 

9128 

9134 

35 

55 

9.9134 

9139 

9144 

9149 

9155 

9160 

9165 

9170 

9175 

9181 

9186 

34 

56 
57 

9.9186 
9.9236 

9191 
9241 

9196 
9246 

9201 
9251 

9206 
9255 

9211 

9260 

9216 
9265 

9221 
9270 

9226 
9275 

9231 
9279 

9236 
9284 

33 
32 

58 

9.9284 

9289 

9294 

9298 

9303 

9308 

9312 

9317 

9322 

9326 

9331 

31 

59 

9.9331 

9335 

9340 

9344 

9349 

9353 

9358 

9362 

9367 

9371 

9375 

30° 

60° 

9.9375 

9380 

9384 

9388 

9393 

9397 

9401 

9406 

9410 

9414 

9418 

29 

61 

9.9418 

9422 

9427 

9431 

9435 

9439 

9443 

9447 

9451 

9455 

9459 

28 

62 

9.9459 

9463. 

9467 

9471 

9475 

9479 

9483 

9487 

9491 

9495 

9499 

27 

63 

9.9499 

9503 

9506 

9510 

9514 

9518 

9522 

9525 

9529 

9533 

9537 

26 

64 

9.9537 

9540 

9544 

9548 

9551 

9555 

9558 

9562 

9566 

9569 

9573 

25 

65 

9.9573 

9576 

9580 

9583 

9587 

9590 

9594 

9597 

9601 

9604 

9607 

24 

66 

9.9607 

9611 

9614 

9617 

9621 

9624 

9627 

9631 

9634 

9637 

9640 

23 

67 

9.9640 

9643 

9647 

9650 

9653 

9656 

9659 

9662 

9666 

9669 

9672 

22 

68 

9.9672 

9675 

9678 

9681 

9684 

9687 

9690 

9693 

9696 

9699 

9702 

21 

69 

9.9702 

9704 

9707 

9710 

9713 

9716 

9719 

9722 

9724 

9727 

9730 

20° 

70° 

9.9730 

9733 

9735 

9738 

9741 

9743 

9746 

9749 

9751 

9754 

9757 

19 

71 

9.9757 

9759 

9762 

9764 

9767 

9770 

9772 

9775 

9777 

9780 

9782 

18 

72 

9.9782 

9785 

9787 

9789 

9792 

9794 

9797 

9799 

9801 

9804 

9806 

17 

73 

9.9806 

9808 

9811 

9813 

9815 

9817 

9820 

9822 

9824 

9826 

9828 

16 

74 

9.9828 

9831 

9833 

9835 

9837 

9839 

9841 

9843 

9845 

9847 

'9849 

15 

75 

9.9849 

9851 

9853 

9855 

9857 

9859 

9861 

9863 

9865 

9867 

9869 

14 

76 

9.9869 

9871 

9873 

9875 

9876 

9878 

9880 

9882 

9884 

9885 

9887 

13 

77 

9.9887 

9889 

9891 

9892 

9894 

9896 

9897 

9899 

9901 

9902 

9904 

12 

78 

9.9904 

9906 

9907 

9909 

9910 

9912 

9913 

9915 

9916 

9918 

9919 

11 

79 

9.9919 

9921 

9922 

9924 

9925 

9927 

9928 

9929 

9931 

9932 

9934 

10° 

80° 

9.9934 

9935 

9936 

9937 

9939 

9940 

9941 

9943 

9944 

9945 

9946 

9 

81 

9.9946 

9947 

9949 

9950 

9951 

9952 

9953 

9954 

9955 

9956 

9958 

8 

82 

9.9958 

9959 

9960 

9961 

9962 

9963 

9964 

9965 

9966 

9967 

9968 

7 

83 

9.9968 

9968 

9969 

9970 

9971 

9972 

9973 

9974 

9975 

9975 

9976 

6 

84 

9.9976 

9977 

9978 

9978 

9979 

9980 

9981 

9981 

9982 

9983 

9983 

5 

85 

9.9983 

9984 

9985 

9985 

9986 

9987 

9987 

9988 

9988 

9989 

9989 

4 

86 

9.9989 

9990 

9990 

9991 

9991 

9992 

9992 

9993 

9993 

9994 

9994 

3 

87 

9.9994 

9994 

9995 

9995 

9996 

9996 

9996 

9996 

9997 

9997 

9997 

2 

88 

9.9997 

9998 

9998 

9998 

9998 

9999 

9999 

9999 

9999 

9999 

9999 

1 

89 

9.9999 

9999 

*0000 

*0000 

*0000 

*0000 

*0000 

*0000 

*0000 

0000 

*0000 

0° 

90° 

0.0000 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

L.  Cos. 

[107] 


L.  Tang. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

«••••••_•_ 

o°.o 

OB^HBMK 

—  00 

6.2419 

«••» 
5429 

7190 

8439 

9408 

*0200 

*0870 

*1450 

*1961 

*2419 

89.9 

0.1 

7.2419 

2833 

3211 

3558 

3880 

4180 

4460 

4723 

4972 

5206 

5429 

89.8 

0.2 

7.5429 

5641 

5843 

6036 

6221 

6398 

6569 

6732 

6890 

7043 

7190 

89.7 

0.3 

7.7190 

7332 

7470 

7604 

7734 

7860 

7982 

8101 

8217 

8329 

8439 

89.6 

0.4 

7.8439 

8547 

8651 

8754 

8853 

8951 

9046 

9140 

9231 

9321 

9409 

89.5 

0.5 

7.9409 

9495 

9579 

9662 

9743 

9823 

9901 

9978 

*0053 

*0127 

*0200 

89.4 

0.6 

8.0200 

0272 

0343 

0412 

0481 

0548 

0614 

0680 

0744 

0807 

087C 

89.3 

0.7 

8.0870 

0932 

0992 

1052 

1111 

1170 

1227 

1284 

1340 

1395 

14  5C 

89.2 

0.8 

8.1450 

1504 

1557 

1610 

1662 

1713 

1764 

1814 

1864 

1913 

1962 

89.1 

0.9 

8.1962 

2010 

2057 

2104 

2150 

2196 

2242 

2287 

2331 

2376 

2419 

89°.0 

1°.0 

8.2419 

2462 

2505 

2548 

2590 

2631 

2672 

2713 

2754 

2794 

2833 

88.9 

1.1 

8.2833 

2873 

2912 

2950 

2988 

3026 

3064 

3101 

3138 

3175 

3211 

88.8 

1.2 

8.3211 

3247 

3283 

3318 

3354 

3389 

3423 

3458 

3492 

3525 

3559 

88.7 

1.3 

8.3559 

3592 

3625 

3658 

3691 

3723 

3755 

3787 

3818 

3850 

3881 

88.6 

1.4 

8.3881 

3912 

3943 

3973 

4003 

4033 

4063 

4093 

4122 

4152 

4181 

88.5 

1.5 

8.4181 

4210 

4238 

4267 

4295 

4323 

4351 

4379 

4406 

4434 

4461 

88.4 

1.6 

8.4461 

4488 

4515 

4542 

4568 

4595 

4621 

4647 

4673 

4699 

4725 

88.3 

1.7 

8.4725 

4750 

4775 

4801 

4826 

4851 

4875 

4900 

4924 

4949 

4973 

88.2 

1.8 

8.4973 

4997 

5021 

5045 

5068 

5092 

5115 

5139 

5162 

5185 

5208 

88.1 

1.9 

8.5208 

5231 

5253 

5276 

5298 

5321 

5343 

5365 

5387 

5409 

5431 

88°.O 

2°.0 

8.5431 

5453 

5474 

5496 

5517 

5538 

5559 

5580 

5601 

5622 

5643 

87.9 

2.1 

8.5643 

5664 

5684 

5705 

5725 

5745 

5765 

5785 

5805 

5825 

5845 

87.8 

2.2 

8.5845 

5865 

5884 

5904 

5923 

5943 

5962 

5981 

6000 

6019 

6038 

87.7 

2.3 

8.6038 

6057 

6076 

6095 

6113 

6132 

6150 

6169 

6187 

6205 

6223 

87.6 

2.4 

8.6223 

6242 

6260 

6277 

6295 

6313 

6331 

6348 

6366 

6384 

6401 

87.5 

2.5 

8.6401 

6418 

6436 

6453 

6470 

6487 

6504 

6521 

6538 

6555 

6571 

87.4 

2.6 

8.6571 

6588 

6605 

6621 

6638 

6654 

6671 

6687 

6703 

6719 

6736 

87.3 

2.7 

8.6736 

6752 

6768 

6784 

6800 

6815 

6831 

6847 

6863 

6878 

6894 

87.2 

2.8 

8.6894 

6909 

6925 

6940 

6956 

6971 

6986 

7001 

7016 

7031 

7046 

87.1 

2.9 

8.7046 

7061 

7076 

7091 

7106 

7121 

7136 

7150 

7165 

7179 

7194 

87°.O 

3°.0 

8.7194 

7208 

7223 

7237 

7252 

7266 

7280 

7294 

7308 

7323 

7337 

86.9 

3.1 

8.7337 

7351 

7365 

7379 

7392 

7406 

7420 

7434 

7448 

7461 

7475 

86.8 

3.2 

8.7475 

7488 

7502 

7515 

7529 

7542 

7556 

7569 

7582 

7596 

7609 

86.7 

3.3 

8.7609 

7622 

7635 

7648 

7661 

7674 

7687 

7700 

7713 

7726 

7739 

86.6 

3.4 

8.7739 

7751 

7764 

7777 

7790 

7802 

7815 

7827 

7840 

7852 

7865 

86.5 

3.5 

8.7865 

7877 

7890 

7902 

7914 

7927 

7939 

7951 

7963 

7975 

7988 

86.4 

3.6 

8.7988 

8000 

8012 

8024 

8036 

8048 

8059 

8071 

8083 

8095 

8107 

86.3 

3.7 

8.8107 

8119 

8130 

8142 

8154 

8165 

8177 

8188 

8200 

8212 

8223 

86.2 

3.8 

8.8223 

8234 

8246 

8257 

8269 

8280 

8291 

8302 

8314 

8325 

8336 

86.1 

3.9 

8.8336 

8347 

8358 

8370 

8381 

8392 

8403 

8414 

8425 

8436 

8446 

86°.0 

4°.0 

8.8446 

8457 

8468 

8479 

8490 

8501 

8511 

8522 

8533 

8543 

8554 

85.9 

4.1 

8.8554 

8565 

8575 

8586 

8596 

8607 

8617 

8628 

8638 

8649 

8659 

85.8 

4.2 

8.8659 

8669 

8680 

8690 

8700 

8711 

8721 

8731 

8741 

8751 

8762 

85.7 

4.3 

8.8762 

8772 

8782 

8792 

8802 

8812 

8822 

8832 

8842 

8852 

8862 

85.6 

4.4 

8.8862 

8872 

8882 

8891 

8901 

8911 

8921 

8931 

8940 

8950 

8960 

85.5 

4.5 

8.8960 

8970 

8979 

8989 

8998 

9008 

9018 

9027 

9037 

9046 

9056 

85.4 

4.6 

8.9056 

9065 

9075 

9084 

9093 

9103 

9112 

9122 

9131 

9140 

9150 

85.3 

4.7 

8.9150 

9159 

9168 

9177 

9186 

9196 

9205 

9214 

9223 

9232 

9241 

85.2 

4.8 

8.9241 

9250 

9260 

9269 

9278 

9287 

9296 

9305 

9313 

9322 

9331 

85.1 

4.9 

8.9331 

9340 

9349 

9358 

9367 

9376 

9384 

9393 

"""•™ 

9402 

9411 

9420 

85°.0 

•••••••^•M 

9 

8 

7 

6 

5 

4 

3 

2 

1 

O 

L.  Cot. 

[108] 


L.  Tang. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

5°.0 

8.9420 

9428 

9437 

9446 

9454 

9463 

9472 

•••••• 

9480 

9489 

9497 

^^•••••i 

9506 

84.9 

5.1 

8.9506 

9515 

9523 

9532 

9540 

9549 

9557 

9565 

9574 

9582 

9591 

84.8 

5.2 

8.9591 

9599 

9608 

9616 

9624 

9633 

9641 

9649 

9657 

9666 

9674 

84.7 

5.3 

8.9674 

9682 

9690 

9699 

9707 

9715 

9723 

9731 

9739 

9747 

9756 

84.6 

5.4 

8.9756 

9764 

9772 

9780 

9788 

9796 

9804 

9812 

9820 

9828 

9836 

84.5 

5.5 

8.9836 

9844 

9852 

9860 

9867 

9875 

9883 

9891 

9899 

9907 

9915 

84.4 

5.6 

8.9915 

9922 

9930 

9938 

9946 

9953 

9961 

9969 

9977 

9984 

9992 

84.3 

5.7 

8.9992 

*0000 

*0007 

*0015 

*0022 

*0030  *0038 

*0045 

*0053 

*0060 

*0068 

84.2 

5.8 

9.0068 

0075 

0083 

0090 

0098 

0105 

0113 

0120 

0128 

0135 

0143 

84.1 

5.9 

9.0143 

0150 

0157 

0165 

0172 

0180 

0187 

0194 

0202 

0209 

0216 

84°.0 

6°.0 

9.0216 

0223 

0231 

0238 

0245 

0253 

0260 

0267 

0274 

0281 

0289 

83.9 

6.1 

9.0289 

0296 

0303 

0310 

0317 

0324 

0331 

0338 

0346 

0353 

0360 

83.8 

6.2 

9.0360 

0367 

0374 

0381 

0388 

0395 

0402 

0409 

0416 

0423 

0430 

83.7 

6.3 

9.0430 

0437 

0444 

0451 

0457 

0464 

0471 

0478 

0485 

0492 

0499 

83.6 

6.4 

9.0499 

0506 

0512 

0519 

0526 

0533 

0540 

0546 

0553 

0560 

0567 

83.5 

6.5 

9.0567 

0573 

0580 

0587 

0593 

0600 

0607 

0614 

0620 

0627 

0633 

83.4 

6.6 

9.0633 

0640 

0647 

0653 

0660 

0667 

0673 

0680 

0686 

0693 

0699 

83.3 

6.7 

9.0699 

0706 

0712 

0719 

0725 

0732 

0738 

0745 

0751 

0758 

0764 

83.2 

6.8 

9.0764 

0771 

0777 

0784 

0790 

0796 

0803 

0809 

0816 

0822 

0828 

83.1 

6.9 

90828 

0835 

0841 

0847 

0854 

0860 

0866 

0873 

0879 

0885 

0891 

83°.0 

7°.0 

9.0891 

0898 

0904 

0910 

0916 

0923 

0929 

0935 

0941 

0947 

0954 

82.9 

7.1 

9.0954 

0960 

0966 

0972 

0978 

0984 

0991 

0997 

1003 

1009 

1015 

82.8 

7.2 

9.1015 

1021 

1027 

1033 

1039 

1045 

1051 

1058 

1064 

1070 

1076 

82.7 

7.3 

9.1076 

1082 

1088 

1094 

1100 

1106 

1112 

1117 

1123 

1129 

1135 

82.6 

7.4 

9.1135 

1141 

1147 

1153 

1159 

1165 

1171 

1177 

1183 

1188 

1194 

82.5 

7.5 

9.1194 

1200 

1206 

1212 

1218 

1223 

1229 

1235 

1241 

1247 

1252 

82.4 

7.6 

9.1252 

1258 

1264 

1270 

1276 

1281 

1287 

1293 

1299 

1304 

1310 

82.3 

7.7 

9.1310 

1316 

1321 

1327 

1333 

1338 

1344 

1350 

1355 

1361 

1367 

82.2 

7.8 

9.1367 

1372 

1378 

1384 

1389 

1395 

1400 

1406 

1412 

1417 

1423 

82.1 

7.9 

9.1423 

1428 

1434 

1439 

1445 

1450 

1456 

1461 

1467 

1473 

1478 

82°.0 

8°.0 

9.1478 

1484 

1489 

1494 

1500 

1505 

1511 

1516 

1522 

1527 

1533 

81.9 

8.1 

9.1533 

1538 

1544 

1549 

1554 

1560 

1565 

1571 

1576 

1581 

1587 

81.8 

8.2 

9.1587 

1592 

1597 

1603 

1608 

1613 

1619 

1624 

1629 

1635 

1640 

81.7 

8.3 

9.1640 

1645 

1651 

1656 

1661 

1667 

1672 

1677 

1682 

1688 

1693 

81.6 

8.4 

9.1693 

1698 

1703 

1709 

1714 

1719 

1724 

1729 

1735 

1740 

1745 

81.5 

8.5 

9.1745 

1750 

1755 

1761 

1766 

1771 

1776 

1781 

1786 

1791 

1797 

81.4 

8.6 

9.1797 

1802 

1807 

1812 

1817 

1822 

1827 

1832 

1837 

1842 

1848 

81.3 

8.7 

9.1848 

1853 

1858 

1863 

1868 

1873 

1878 

1883 

1888 

1893 

1898 

81.2 

8.8 

9.1898 

1903 

1908 

1913 

1918 

1923 

1928 

1933 

1938 

1943 

1948 

81.1 

8.9 

9.1948 

1953 

1958 

1963 

1968 

1973 

1977 

1982 

1987 

1992 

1997 

81°.0 

9°.0 

9.1997 

2002 

2007 

2012 

2017 

2022 

2026 

2031 

2036 

2041 

2046 

80.9 

9.1 

9.2046 

2051 

2056 

2060 

2065 

2070 

2075 

2080 

2085 

2089 

2094 

80.8 

9.2 

9.2094 

2099 

2104 

2109 

2113 

2118 

2123 

2128 

2132 

2137 

2142 

80.7 

9.3 

9.2142 

2147 

2151 

2156 

2161 

2166 

2170 

2175 

2180 

2185 

2189 

80.6 

9.4 

9.2189 

2194 

2199 

2203 

2208 

2213 

2217 

2222 

2227 

2231 

2236 

80.5 

9.5 

9.2236 

2241 

2245 

2250 

2255 

2259 

2264 

2269 

2273 

2278 

2282 

80.4 

9.6 

9.2282 

2287 

2292 

2296 

2301 

2305 

2310 

2315 

2319 

2324 

2328 

80.3 

.  9.7 

9.2328 

2333 

2337 

2342 

2346 

2351 

2356 

2360 

2365 

2369 

2374 

80.2 

9.8 

9.2374 

2378 

2383 

2387 

2392 

2396 

2401 

2405 

2410 

2414 

2419 

80.1 

9.9 

9.2419 

IMHI^HMHM 

2423 

2428 

2432 

2437 

2441 

2445 

2450 

2454 

2459 

2463 

80°.0 

9 

8 

7 

6 

5 

4 

3 

2 

1 

O 

L.  Cot. 

[109] 


iL.Tang. 
I™  ^^^^ 

0 

1 

2 

5 

6 

7 

8 

9 

1 

mmmmmmmmim 

«"•"• 

^BBHBBHHI 

~ 

•^^ 

•— 

-oo 

90° 

0° 

—  oo 

7.2419 

5429 

7190 

8439 

9409 

*0200 

*0870 

*1450 

*1962 

*2419 

89 

1 

8.2419 

2833 

3211 

3559 

3881 

4181 

4461 

4725 

4973 

5208 

5431 

88 

2 

8.5431 

5643 

5845 

6038 

6223 

6401 

6571 

6736 

6894 

7046 

7194 

87 

3 

8.7194 

7337 

7475 

7609 

7739 

7865 

7988 

8107 

8223 

8336 

8446 

86 

4 

8.8446 

8554 

8659 

8762 

8862 

8960 

9056 

9150 

9241 

9331 

9420 

85 

5 

8.9420 

9506 

9591 

9674 

9756 

9836 

9915 

9992 

*0068 

*0143 

*0216 

84 

6 

9.0216 

0289 

0360 

0430 

0499 

0567 

0633 

0699 

0764 

0828 

0891 

83 

7 

9.0891 

0954 

1015 

1076 

1135 

1194 

1252 

1310 

1367 

1423 

1478 

82 

8 

9.1478 

1533 

1587 

1640 

1693 

1745 

1797 

1848 

1898 

1948 

1997 

81 

9 

9.1997 

2046 

2094 

2142 

2189 

2236 

2282 

2328 

2374 

2419 

2463 

80° 

10° 

9.2463 

2507 

2551 

2594 

2637 

2680 

2722 

2764 

2805 

2846 

2887 

79 

11 

9.2887 

2927 

2967 

3006 

3046 

3085 

3123 

3162 

3200 

3237 

3275 

78 

12 

9.3275 

3312 

3349 

3385 

3422 

3458 

3493 

3529 

3564 

3599 

3634 

77 

13 

9.3634 

3668 

3702 

3736 

3770 

3804 

3837 

3870 

3903 

3935 

3968 

76 

14 

9.3968 

4000 

4032 

4064 

4095 

4127 

4158 

4189 

4220 

4250 

4281 

75 

15 

9.4281 

4311 

4341 

4371 

4400 

4430 

4459 

4488 

4517 

4546 

4575 

74 

16 

9.4575 

4603 

4632 

4660 

4688 

4716 

4744 

4771 

4799 

4826 

4853 

73 

17 

9.4853 

4880 

4907 

4934 

4961 

4987 

5014 

5040 

5066 

5092 

5118 

72 

18 

9.5118 

5143 

5169 

5195 

5220 

5245 

5270 

5295 

5320 

5345 

5370 

71 

19 

9.5370 

5394 

5419 

5443 

5467 

5491 

5516 

5539 

5563 

5587 

5611 

70° 

20° 

9.5611 

5634 

5658 

5681 

5704 

5727 

5750 

5773 

5796 

5819 

5842 

69 

21 

9.5842 

5864 

5887 

5909 

5932 

5954 

5976 

5998 

6020 

6042 

6064 

68 

22 

9.6064 

6086 

6108 

6129 

6151 

6172 

6194 

6215 

6236 

6257 

6279 

67 

23 

9.6279 

6300 

6321 

6341 

6362 

6383 

6404 

6424 

6445 

6465 

6486 

66 

24 

9.6486 

6506 

6527 

6547 

6567 

6587 

6607 

6627 

6647 

6667 

6687 

65 

25 

9.6687 

6706 

6726 

6746 

6765 

6785 

6804 

6824 

6843 

6863 

6882 

64 

26 

9.6882 

6901 

6920 

6939 

6958 

6977 

6996 

7015 

7034 

7053 

7072 

63 

27 

9.7072 

7090 

7109 

7128 

7146 

7165 

7183 

7202 

7220 

7238 

7257 

62 

28 

9.7257 

7275 

7293 

7311 

7330 

7348 

7366 

7384 

7402 

7420 

7438 

61 

29 

9.7438 

7455 

7473 

7491 

7509 

7526 

7544 

7562 

7579 

7597 

7614 

60° 

30° 

9.7614 

7632 

7649 

7667 

7684 

7701 

7719 

7736 

7753 

7771 

7788 

59 

31 

9.778*8 

7805 

7822 

7839 

7856 

7873 

7890 

7907 

7924 

7941 

7958 

58 

32 

9.7958 

7975 

7992 

8008 

8025 

8042 

8059 

8075 

8092 

8109 

8125 

57 

33 

9.8125 

8142 

8158 

8175 

8191 

8208 

8224 

8241 

8257 

8274 

8290 

56 

34 

9.8290 

8306 

8323 

8339 

8355 

8371 

8388 

8404 

8420 

8436 

8452 

55 

35 

9.8452 

8468 

8484 

8501 

8517 

8533 

8549 

8565 

8581 

8597 

8613 

54 

36 

9.8613 

8629 

8644 

8660 

8676 

8692 

8708 

8724 

8740 

8755 

8771 

53 

37 

9.8771 

8787 

8803 

8818 

8834 

8850 

8865 

8881 

8897 

8912 

8928 

52 

38 

9.8928 

8944 

8959 

8975 

8990 

9006 

9022 

9037 

9053 

9068 

9084 

51 

39 

9.9084 

9099 

9115 

9130 

9146 

9161 

9176 

9192 

9207 

9223 

9238 

50° 

40° 

9.9238 

9254 

9269 

9284 

9300 

9315 

9330 

9346 

9361 

9376 

9392 

49 

41 

9.9392 

9407 

9422 

9438 

9453 

9468 

9483 

9499 

9514 

9529 

9544 

48 

42 

9.9544 

9560 

9575 

9590 

9605 

9621 

9636 

9651 

9666 

9681 

9697 

47 

43 

9.9697 

9712 

9727 

9742 

9757 

9772 

9788 

9803 

9818 

9833 

9848 

46 

44 

9.9848 

9864 

9879 

9894 

9909 

9924 

9939 

9955 

9970 

9985 

*0000 

45° 

45° 

0.0000 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

L.  Cot. 

[110] 


L.  Tang. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0.0000 

45° 

45° 

0.0000 

0015 

0030 

0045 

0061 

0076 

0091 

0106 

0121 

0136 

0152 

44 

46 

0152 

0167 

0182 

0197 

0212 

0228 

0243 

0258 

0273 

0288 

0303 

43 

47 

0303 

0319 

0334 

0349 

0364 

0379 

0395 

0410 

0425 

0440 

0456 

42 

48 

0456 

0471 

0486 

0501 

0517 

0532 

0547 

0562 

0578 

0593 

0608 

41 

49 

0608 

0624 

0639 

0654 

0670 

0685 

0700 

0716 

0731 

0746 

0762 

40° 

50° 

0.0762 

0777 

0793 

0808 

0824 

0839 

0854 

0870 

0885 

0901 

0916 

39 

51 

0916 

0932 

0947 

0963 

0978 

0994 

1010 

1025 

1041 

1056 

1072 

38 

52 

1072 

1088 

1103 

1119 

1135 

1150 

1166 

1182 

1197 

1213 

1229 

37 

53 

1229 

1245 

1260 

1276 

1292 

1308 

1324 

1340 

1356 

1371 

1387 

36 

54 

1387 

1403 

1419 

1435 

1451 

1467 

1483 

1499 

1516 

1532 

1548 

35 

55 

1548 

1564 

1580 

1596 

1612 

1629 

1645 

1661 

1677 

1694 

1710 

34 

56 

1710 

1726 

1743 

1759 

1776 

1792 

1809 

1825 

1842 

1858 

1875 

33 

57 

1875 

1891 

1908 

1925 

1941 

1958 

1975 

1992 

2008 

2025 

2042 

32 

58 

2042 

2059 

2076 

2093 

2110 

2127 

2144 

2161 

2178 

2195 

2212 

31 

59 

2212 

2229 

2247 

2264 

2281 

2299 

2316 

2333 

2351 

2368 

2386 

30° 

60° 

0.2386 

2403 

2421 

2438 

2456 

2474 

2491 

2509 

2527 

2545 

2562 

29 

61 

2562 

2580 

2598 

2616 

2634 

2652 

2670 

2689 

2707 

2725 

2743 

28 

62 

2743 

2762 

2780 

2798 

2817 

2835 

2854 

2872 

2891 

2910 

2928 

27 

63 

2928 

2947 

2966 

2985 

3004 

3023 

3042 

3061 

3080 

3099 

3118 

26 

64 

3118 

3137 

3157 

3176 

3196 

3215 

3235 

3254 

3274 

3294 

3313 

25 

65 

3313 

3333 

3353 

3373 

3393 

3413 

3433 

3453 

3473 

3494 

3514 

24 

66 

3514 

3535 

3555 

3576 

3596 

3617 

3638 

3659 

3679 

3700 

3721 

23 

67 

3721 

3743 

3764 

3785 

3806 

3828 

3849 

3871 

3892 

3914 

3936 

22 

68 

3936 

3958 

3980 

4002 

4024 

4046 

4068 

4091 

4113 

4136 

4158 

21 

69 

4158 

4181 

4204 

4227 

4250 

4273 

4296 

4319 

4342 

4366 

4389 

20° 

70° 

0.4389 

4413 

4437 

4461 

4484 

4509 

4533 

4557 

45«1 

4606 

4630 

19 

71 

4630 

4655 

4680 

4705 

4730 

4755 

4780 

4805 

4831 

4857 

4882 

18 

72 

4882 

4908 

4934 

4960 

4986 

5013 

5039 

5066 

5093 

5120 

5147 

17 

73 

5147 

5174 

5201 

5229 

5256 

5284 

5312 

5340 

5368 

5397 

5425 

16 

74 

5425 

5454 

5483 

5512 

5541 

5570 

5600 

5629 

5659 

5689 

5719 

15 

75 

5719 

5750 

5780 

5811 

5842 

5873 

5905 

5936 

5968 

6000 

6032 

14 

76 

6032 

6065 

6097 

6130, 

6163 

6196 

6230 

6264 

6298 

6332 

6366 

13 

77 

6366 

6401 

6436 

6471 

6507 

6542 

6578 

6615 

6651 

6688 

6725 

12 

78 

6725 

6763 

6800 

6838 

6877 

6915 

6954 

6994 

7033 

7073 

7113 

11 

79 

7113 

7154 

7195 

7236 

7278 

7320 

7363 

7406 

7449 

7493 

7537 

10° 

80° 

0.7537 

7581 

7626 

7672 

7718 

7764 

7811 

7858 

7906 

7954 

8003 

9 

81 

8003 

8052 

8102 

8152 

8203 

8255 

8307 

8360 

8413 

8467 

8522 

8 

82 

8522 

8577 

8633 

8690 

8748 

8806 

8865 

8924 

8985 

9046 

9109 

7 

83 

9109 

9172 

9236 

9301 

9367 

9433 

9501 

9570 

9640 

9711 

9784 

6 

84 

0.9784 

9857 

9932 

*0008 

*0085 

*0164 

*0244 

*0326 

*0409 

*0494 

*0580 

5 

• 

85 

1.0580 

0669 

0759 

0850 

0944 

1040 

1138 

1238 

1341 

1446 

1554 

4 

86 

1554 

1664 

1777 

1893 

2012 

2135 

2261 

2391 

2525 

2663 

2806 

3 

87 

2806 

2954 

3106 

3264 

3429 

3599 

3777 

3962 

4155 

4357 

4569 

2 

88 

4569 

4792 

5027 

5275 

5539 

5819 

6119 

6441 

6789 

7167 

7581 

1 

89 

1.7581 

8038 

8550 

9130 

9800 

*0591 

*1561 

*2810 

*4571 

*7581 

00 

0° 

90° 

CO 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

L.  Cot. 

[Ill] 


L.  Tang. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

80°.0 

0.7537 

7541 

7546 

7550 

7555 

7559 

7563 

7568 

7572 

7577 

7581 

9.9 

80.1 

7581 

7586 

7590 

7595 

7599 

7604 

7608 

7613 

7617 

7622 

7626 

9.8 

80.2 

7626 

7631 

7635 

7640 

7644 

7649 

7654 

7658 

7663 

7667 

7672 

9.7 

80.3 

7672 

7676 

7681 

7685 

7690 

7695 

7699 

7704 

7708 

7713 

7718 

9.6 

80.4 

7718 

7722 

7727 

7731 

7736 

7741 

7745 

7750 

7755 

7759 

7764 

9.5 

80.5 

7764 

7769 

7773 

7278 

7783 

7787 

7792 

7797 

7801 

7806 

7811 

9.4 

80.6 

7811 

7815 

7820 

7825 

7830 

7834 

7839 

7844 

7849 

7853 

7858 

9.3 

80.7 

7858 

7863 

7868 

7872 

7877 

7882 

7887 

7891 

7896 

7901 

7906 

9.2 

80.8 

7906 

7911 

7915 

7920 

7925 

7930 

7935 

7940 

7944 

7949 

7954 

9.1 

80.9 

7954 

7959 

7964 

7969 

7974 

7978 

7983 

7988 

7993 

7998 

8003 

9°.0 

81°.0 

0.8003 

8008 

8013 

8018 

8023 

8027 

8032 

8037 

8042 

8047 

8052 

8.9 

81.1 

8052 

8057 

8062 

8067 

8072 

8077 

8082 

8087 

8092 

8097 

8102 

8.8 

81.2 

8102 

8107 

8112 

8117 

8122 

8127 

8132 

8137 

8142 

8147 

8152 

8.7 

81.3 

8152 

8158 

8163 

8168 

8173 

8178 

8183 

8188 

8193 

8198 

8203 

8.6 

81.4 

8203 

8209 

8214 

8219 

8224 

8229 

8234 

8239 

8245 

8250 

8255 

8.5 

81.5 

8255 

8260 

8265 

8271 

8276 

8281 

8286 

8291 

8297 

8302 

8307 

8.4 

81.6 

8307 

8312 

8318 

8323 

8328 

8333 

8339 

8344 

8349 

8355 

8360 

8.3 

81.7 

8360 

8365 

8371 

8376 

8381 

8387 

8392 

8397 

8403 

8408 

8413 

8.2 

81.8 

8413 

8419 

8424 

8429 

8435 

8440 

8446 

8451 

8456 

8462 

8467 

8.1 

81.9 

8467 

8473 

8478 

8484 

8489 

8495 

8500 

8506 

8511 

8516 

8522 

8°.0 

82°.0 

0.8522 

8527 

8533 

8539 

8544 

8550 

8555 

8561 

8566 

8572 

8577 

7.9 

82.1 

8577 

8583 

8588 

8594 

8600 

8605 

8611 

8616 

8622 

8628 

8633 

7.8 

82.2 

8633 

8639 

8645 

8650 

6856 

8662 

8667 

8673 

8679 

8684 

8690 

7.7 

82.3. 

8690 

8696 

8701 

8707 

8713 

8719 

8724 

8730 

8736 

8742 

8748 

7.6 

82.4 

8748 

8753 

8759 

8765 

8771 

8777 

8782 

8788 

8794 

8800 

8806 

7.5  . 

82.5 

8806 

8812 

8817 

8823 

8829 

8835 

8841 

8847 

8853 

8859 

8865 

7.4 

82.6 

8865 

8871 

8877 

8883 

8888 

8894 

8900 

8906 

8912 

8918 

8924 

7.3 

82.7 

8924 

8930 

8936 

8942 

8949 

8955 

8961 

8967 

8973 

8979 

8985 

7.2 

82.8 

8985 

8991 

8997 

9003 

9009 

9016 

9022 

9028 

9034 

9040 

9046 

7.1 

82.9 

9046 

9053 

9059 

9065 

9071 

9077 

9084 

9090 

9096 

9102 

9109 

7°.0 

83°.0 

0.9109 

9115 

9121 

9127 

9134 

9140 

9146 

9153 

9159 

9165 

9172 

6.9 

83.1 

9172 

9178 

9184 

9191 

9197 

9204 

9210 

9216 

9223 

9229 

9236 

6.8 

83.2 

9236 

9242 

9249 

9255 

9262 

9268 

9275 

9281 

9288 

9294 

9301 

6.7 

83.3 

9301 

9307 

9314 

9320 

9327 

9333 

9340 

9347 

9353 

9360 

9367 

6.6 

83.4 

9367 

9373 

9380 

9386 

9393 

9400 

9407 

9413 

9420 

9427 

9433 

6.5 

83.5 

9433 

9440 

9447 

9454 

9460 

9467 

9474 

9481 

9488 

9494 

9501 

6.4 

83.6 

9501 

9508 

9515 

9522 

•9529 

9536 

9543 

9549 

9556 

9563 

9570 

6.3 

83.7 

9570 

9577 

9584 

9591 

9598 

9605 

9612 

9619 

9626 

9633 

9640 

6.2 

83.8 

9640 

9647 

9654 

9662 

9669 

9676 

9683 

9690 

9697 

9704 

9711 

6.1 

83.9 

9711 

9719 

9726 

9733 

9740 

9747 

9755 

9762 

9769 

9777 

9784 

6°.O 

84°.0 

0.9784 

9791 

9798 

9806 

9813 

9820 

9828 

9835 

9843 

9850 

9857 

5.9 

84.1 

9857 

9865 

9872 

9880 

9887 

9895 

9902 

9910 

9917 

9925 

9932 

5.8 

84.2 

0.9932 

9940 

9947 

9955 

9962 

9970 

9978 

9985 

9993 

*0000 

*0008 

5.7 

84.3 

1.0008 

0016 

0023 

0031 

'0039 

0047 

0054 

0062 

0070 

0078 

0085 

•6.6 

84.4 

0085 

0093 

0101 

0109 

0117 

0125 

0133 

0140 

0148 

0156 

0164 

5.5 

84.5 

0164 

0172 

0180 

0188 

0196 

0204 

0212 

0220 

0228 

0236 

0244 

5.4 

84.6 

0244 

0253 

0261 

0269 

0277 

0285 

0293 

0301 

0310 

0318 

0326 

5.3 

84.7 

0326 

0334 

0343 

0351 

0359 

0367 

0376 

0384 

0392 

0401 

0409 

5.2 

84.8 

0409 

0418 

0426 

0435 

0443 

0451 

0460 

0468 

0477 

0485 

0494 

5.1 

84.9 

1.0494 

0503 

0511 

0520 

0528 

0537 

0546 

0554 

0563 

0572 

0580 

5°.O 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

L.  Cot. 

[112] 


L.  Taiig. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

85°.0 

1.0580 

0589 

0598 

0607 

0616 

0624 

0633 

0642 

0651 

0660 

0669 

4.9 

85.1 

0669 

0678 

0687 

0695 

0704 

0713 

0722 

0731 

0740 

0750 

0759 

4.8 

85.2 

0759 

0768 

0777 

0786 

0795 

0804 

0814 

0823 

0832 

0841 

0850 

4.7 

85.3 

0850 

0860 

0869 

0878 

0888 

0897 

0907 

0916 

0925 

0935 

0944 

4.6 

85.4 

0944 

0954 

0963 

0973 

0982 

0992 

1002 

1011 

1021 

1030 

1040 

4.5 

85.5 

1040 

1050 

1060 

1069 

1079 

1089 

1099 

1109 

1118 

1128 

1138 

4.4 

85.6 

1138 

1148 

1158 

1168 

1178 

1188 

1198 

1208 

1218 

1228 

1238 

4.3 

85.7 

1238 

1249 

1259 

1269 

1279 

1289 

1300 

1310 

1320 

1331 

1341 

4.2 

85.8 

1341 

1351 

1362 

1372 

1383 

1393 

1404 

1414 

1425 

1435 

1446 

4.1 

85.9 

1446 

1457 

1467 

1478 

1489 

1499 

1510 

1521 

1532 

1543 

1554 

4°.0 

86°.0 

1.1554 

1564 

1575 

1586 

1597 

1608 

1619 

1630 

1642 

1653 

1664 

3.9 

86.1 

1664 

1675 

1686 

1698 

1709 

1720 

1731 

1743 

1754 

1766 

1777 

3.8 

86.2 

1777 

1788 

1800 

1812 

1823 

1835 

1846 

1858 

1870 

1881 

1893 

3.7 

86.3 

1893 

1905 

1917 

1929 

1941 

1952 

1964 

1976 

1988 

2000 

2012 

3.6 

86.4 

2012 

2025 

2037 

2049 

2061 

2073 

2086 

2098 

2110 

2123 

2135 

3.5 

86.5 

2135 

2148 

2160 

2173 

2185 

2198 

2210 

2223 

2236 

2249 

2261 

3.4 

86.6 

2261 

2274 

2280 

2300 

2313 

2326 

2339 

2352 

2365 

2378 

2391 

3.3 

86.7 

2391 

2404 

2418 

2431 

2444 

2458 

2471 

2485 

2498 

2512 

2525 

3.2 

86.8 

2525 

2539 

2552 

2566 

2580 

2594 

2608 

2621 

2635 

2649 

2663 

3.1 

86.9 

2663 

2677 

2692 

2706 

2720 

2734 

2748 

2763 

2777 

2792 

2806 

3°.0 

87°.0 

1.2806 

2821 

2835 

2850 

2864 

2879 

2894 

2909 

2924 

2939 

2954 

2.9 

87.1 

2954 

2969 

2984 

2999 

3014 

3029 

3044 

30GO 

3075 

3091 

3106 

2.8 

87.2 

3106 

3122 

3137 

3153 

3169 

3185 

3200 

3216 

3232 

3248 

3264 

2.7 

87.3 

3264 

3281 

3297 

3313 

3329 

3346 

3362 

3379 

3395 

3412 

3429 

2.6 

87.4 

3429 

3445 

3462 

3479 

3496 

3513 

3530 

3547 

3564 

3582 

3599 

2.5 

87.5 

3599 

3616 

3634 

3652 

3669 

3687 

3705 

3723 

3740 

3758 

3777 

2.4 

87.6 

3777 

3795 

3813 

3831 

3850 

3868 

3887 

3905 

3924 

3943 

3962 

2.3 

87.7 

3962 

3981 

4000 

4019 

4038 

4057 

4077 

4096 

4116 

4135 

4155 

2.2 

87.8 

4155 

4175 

4195 

4215 

4235 

4255 

4275 

4295 

4316 

4336 

4357 

2.1 

87.9 

4357 

4378 

4399 

4420 

4441 

4462 

4483 

45CH 

4526 

4547 

4569 

2°.0 

88°.0 

1.4569 

4591 

4613 

4635 

4657 

4679 

4702 

4724 

4747 

4769 

4792 

1.9 

88.1 

4792 

4815 

4838 

4861 

4885 

4908 

4932 

4955 

4979 

5003 

5027 

1.8 

88.2 

5027 

5051 

5076 

5100 

5125 

5149 

5174 

5199 

5225 

5250 

5275 

1.7 

88.3 

5275 

5301 

5327 

5353 

5379 

5405 

5432 

5458 

5485 

5512 

5539 

1.6 

88.4 

5539 

5566 

5594 

5621 

5649 

5677 

5705 

5733 

5762 

5790 

5819 

1.5 

88.5 

5819 

5848 

5878 

5907 

5937 

5967 

5997 

6027 

6057 

6088 

6119 

1.4 

88.6 

6119 

6150 

6182 

6213 

6245 

6277 

6309 

6342 

6375 

6408 

6441 

1.3 

88.7 

6441 

6475 

6508 

6542 

6577 

6611 

664  6 

6682 

6717 

6753 

6789 

1.2 

88.8 

6789 

6825 

6862 

6899 

6936 

6974 

7012 

7050 

7088 

7127 

7167 

1.1 

88.9 

7167 

7206 

7246 

7287 

7328 

7369 

7410 

7452 

7495 

7538 

7581 

1°.O 

89°.0 

1.7581 

7624 

7669 

7713 

7758 

7804 

7850 

7896 

7943 

7990 

8038 

0.9 

89.1 

8038 

8087 

8136 

8186 

8236 

8287 

8338 

8390 

8443 

8496 

8550 

0.8 

89.2 

8550 

8605 

8660 

8716 

8773 

8830 

8889 

8948 

9008 

9068 

9130 

0.7 

89.3 

9130 

9193 

9256 

9320 

9386 

9452  9519 

9588 

9657 

9728 

9800 

0.6 

89.4 

1.9800 

9873 

9947 

*0022 

*0099 

*0177 

*0257 

*0338 

*0421 

*0505 

*0591 

0.5 

89.5 

2.0591 

0679 

0769 

0860 

0954 

1049 

1147 

1246 

1349 

1453 

1561 

0.4 

89.6 

1561 

1671 

1783 

1899 

2018 

2140 

2266 

2396 

2530 

2668 

2810 

0.3 

89.7 

2810 

2957 

3110 

3268 

3431 

3602 

3779 

3964 

4157 

4359 

4571 

0.2 

89.8 

4571 

4794 

5028 

5277 

5540 

5820 

6120 

6442 

6789 

7167 

7581 

0.1 

89.9 

2.7581 

8039 

8550 

9130 

9800 

*0592 

*1561 

*2810 

*4573 

*7581 

-00 

o°.o 

9 

— 

__ 
6 

5 

4 

-- 

— 

L.  Cot.  1 

[113] 


TABLE  VIII 

CONVERSION  OF  f  "  INTO  DECIMAL  PARTS  OF  A  DEGREE 


V 

0.016° 

11' 

0.183° 

21' 

0.350° 

31' 

0.516° 

41' 

0.683° 

51' 

0.850° 

2' 

.033 

12' 

.200 

22' 

.366 

32' 

.533 

42' 

.700 

52' 

.866 

3' 

.050 

13' 

.216 

23' 

.383 

33' 

.550 

43' 

.716 

53' 

.883 

V 

.066 

14' 

.233 

24' 

.400 

34' 

.566 

44' 

.733 

54' 

.900 

5' 

.083 

15' 

.250 

25' 

.416 

35' 

.583 

45' 

.750 

55' 

.916 

6' 

.100 

16' 

.266 

26' 

.433 

36' 

.600 

46' 

.766 

56' 

.933 

7' 

.116 

17' 

.283 

27' 

.450 

37' 

.616 

47' 

.783 

57' 

.950 

8' 

.133 

18' 

.300 

28' 

.466 

38' 

.633 

48' 

.800 

58' 

.966 

9' 

.150 

19' 

.316 

29' 

.483 

39' 

.650 

49' 

.816 

59' 

.983 

10' 

.166 

20' 

.333 

30' 

.500 

40' 

.666 

50' 

.833 

60' 

1.000 

1" 

0.00028° 

6" 

0.00166° 

10" 

0.00277° 

2" 

.00056 

7" 

.00194 

20" 

.00555 

3" 

.00083 

8" 

.00222 

30" 

.00833 

4" 

.00111 

9" 

.00250 

40" 

.01111 

5" 

.00138 

50" 

.01388 

TABLE   IX 

CONVERSION  OF  DECIMAL  PARTS  OF  A  DEGREE  INTO  f  " 


0.01° 

0'  36" 

0.11° 

6'  36" 

0.21° 

12'  36" 

0.31° 

18'  36" 

.02     . 

1'  12" 

.12 

7'  12" 

.22 

13'  12" 

.32 

19'  12" 

.03 

1'  48" 

.13 

7'  48" 

.23 

13'  48" 

.33 

19'  48" 

.04 

2'  24" 

.14 

8'  24" 

.24 

14'  24" 

.34 

20'  24" 

.05 

3' 

.15 

9' 

.25 

15' 

.35 

21' 

.06 

3'  36" 

.16 

9'  36" 

.26 

15'  36" 

.36 

21'  36" 

.07 

4'  12" 

.17 

10'  12" 

.27 

16'  12" 

.37 

22'  12" 

.08 

4'  48" 

.18 

10'  48" 

.28 

16'  48" 

.38 

22'  48" 

.09 

5'  24" 

.19 

11'  24" 

.29 

17'  24" 

.39 

23'  24" 

.10 

6' 

.20 

12' 

.30 

18' 

.40 

24' 

0.41° 

24'  36" 

0.51° 

30'  36" 

0.61° 

36'  36" 

0.71° 

42'  36" 

.42 

25'  12" 

.52 

31'  12" 

.62 

37'  12" 

.72 

43'  12" 

.43 

25'  48" 

.53 

31'  48" 

.63 

37'  48" 

.73 

43'  48" 

.44 

26'  24" 

.54 

32'  24" 

.64 

38'  24" 

.74 

44'  24" 

.45 

27' 

.55 

33' 

.65 

39' 

.75 

45' 

.46 

27'  36" 

.56 

33'  36" 

.66 

39'  36" 

.76 

45'  36" 

.47 

28'  12" 

.57 

34'  12" 

.67 

40'  12" 

.77 

46'  12" 

.48 

28'  48" 

.58 

34'  48" 

.68 

40'  48" 

.78 

46'  48" 

.49 

29'  24" 

.59 

35'  24" 

.69 

41'  24" 

.79 

47'  24" 

.50 

30' 

.60 

36' 

.70 

42' 

.80 

48' 

0.81° 

48'  36" 

0.91° 

54'  36" 

0.001° 

3.6" 

.82 

49'  12" 

.92 

55'  12" 

.002 

7.2" 

.83 

49'  48" 

.93 

55'  48" 

.003 

10.8" 

.84 

50'  24" 

.94 

56'  24" 

.004 

14.4" 

.85 

51' 

.95 

57' 

.005 

18    " 

.86 

51'  36" 

.96 

57'  36" 

.006 

21.6" 

.87 

52'  12" 

.97 

58'  12" 

.007 

25.2" 

.88 

52'  48" 

.98 

58'  48" 

.008 

28.8" 

.89 

53'  24" 

.99 

59'  24" 

.009 

32.4" 

.90 

54' 

1.00 

60' 

[114] 


ANSWERS 


Exercise  1 

1 .  loga  9  =  2.          logs  27  =  3.         Iog4  64  =  4. 

=  —  4.     logic  £>  =  —  1.     log™  .01  =  -  2. 

2.  Iog2  32  =  5.      log2lfe=-5.       Iog48  =  |. 


3.    1. 


9.    4/64=4. 


logio  .001  =  —  3. 


^4096  = 


logs  |  =  -2. 


Iog8  16  = 


Exercise  2 


1.   2.         3.   2. 

5.  0.             7.   0.          9.    -  3.        11.  0. 

13.    -4.       15.    1. 

2.   4.        4.   1. 

6.    -2.         8.   0.         10.    -5.         12.   3. 

14.   2. 

16.   3  =  4.     2  =  3. 

5  =  6.       1=2.      0  =  1.      4  =  5.      8-10 

=  1.      7-10  =  2. 

9  -  10  =  0. 

Exercise  3 

1.    3.88235. 

8.    1.82751.                 15.    1.93952. 

22.    8.27135-10. 

2.   3.82737. 

9.   0.52410.                  16.    9.88081  -  10. 

23.    4.51427. 

3.   1.91381. 

10.    7.82737-10.         17.   6.09691-10. 

24.    3.51427. 

4.    3.89553. 

11.   4.84510-10.         18.   2.00109. 

25.   2.51427. 

5.    1.87506. 

12.    5.60206.                  19.    1.24622. 

26.    1.51427. 

6.    2.19590. 

13.    1.16505-10.         20.    1.62325. 

27.    0.51427. 

7.   4.55965. 

14.   7.35550.                 21.    4.0000-10. 

28.    log  200  =  2.  30103.        log  3000  =  3.47712.        log  50  =  1.69897.         log!007r  = 

2.49715.           log  20  = 

1  1.30103.            log  .002  =  7.30103  -  10. 

log  30  =  1.47712. 

log  .0005  =  6.69897  - 

10.             log  —  =  8.49715  -  10.             log 
luu 

.3=9.47712-10. 

log  .2  =  9.30103  -  10. 

log  10  TT  =  1.49715.     log  20000  =  4.30103. 

29.    1.1028. 

35.    .0011.                      40.   2.9847. 

45.   4.4619. 

30.   2.8824. 

36.    1.3923.                   41.   0.1666. 

46.    1.2916. 

31.    1.6302. 

37.   9.0459-10.          42.    0.2462. 

47.   9.9358  -  10. 

32.    .0887. 

38.    1.0676.                   43.   5.5655-10. 

48.   8.0012  -  10. 

33.    8.4200  —  10. 

39.    7.1030-10.          44.   7.4213-10. 

49.   0.3474. 

34.   7.1030-  10. 

Exercise  4 

1.   26.22. 

11.  221.705.                 20.   25.6. 

29.  454.44. 

2.    157.6. 

12.    .01569.                    21.   541. 

30.   .0000022337. 

3.   9.627. 

13.   10.88375.                22.   1712. 

31.    657.166. 

4.   48323333.3. 

14.   .50742.                    23.   .14277. 

32.    201.409. 

5.   .16719. 

15.   1647.3.                   24.    107.8. 

33.    .3625. 

6.    .00026827. 

16.    1008581.4.              25.    10.315. 

34.   9.6968-10. 

7.    3896545.45. 

17.    .78488.                   26.   .0106725. 

35.    3.1443. 

8.   .000055855. 

18.   96988.                     27.    .001309. 

36.    49.25. 

9.   100925581.4. 

19.    .69781.                   28.   .000010044. 

37.    .2285. 

10.   .37029. 

4 

ANSWERS 

Exercise 

5 

1.    53295. 

4.   8.3552. 

7.    1.492. 

10.    .96518. 

13.    -.34526. 

2.    1383.62. 

5.    514.055. 

8.    .01141. 

11.  -1.8583. 

14.    $33945. 

3.   211820. 

6.    19.033913. 

9.    5.3921. 

12.   -  .059439. 

15.    $491.04. 

200        ,oi  c 

|i             100  *"      5  4165 

300  x  500      47?46  67 

376 

58      . 

7T 

18.    1.3774  A., 

3.4435  A., 

45.9134  A. 

19.  33.38.      21. 

.4171.        23.  3261.           25. 

3.908.     27.  .0939. 

31.  $325.60. 

20.  6.727.      22. 

2034.3.      24.  1. 

16467.      26. 

3.413.     30.  $213.47. 

32.  $5874.75. 

Exercise 

6 

1.    .972. 

9. 

2.34667. 

19.    .11069. 

29.   6080000. 

2.    99.266. 

10. 

-  .0447. 

20.    2519.6. 

30.    4.245. 

3.    8.9254. 

11. 

-  1.5793. 

21.    7061.67. 

31.    17.49. 

4.    .182916. 

12. 

24.1394. 

22.    65.97  =  66  yr. 

32.   1.272. 

5.    1602.4 

13. 

19.85. 

23.    .5342. 

33.   .4163. 

6.    2.37242. 

14. 

24.035 

24.    1.6167. 

34.    12.07. 

7.    218.51. 

15. 

189.66. 

25.    1.1377. 

35.    5.77. 

6.6943. 

16. 

.12246. 

26.    22.33. 

36.   2316.8. 

7.1845. 

17. 

13306.06. 

27.    10695. 

8.    500  m.  =  1640.5  ft. 
7294  m.  =  23931.11  ft. 

1029.4. 

28.    .1705. 

300  m.  =  984.26  ft. 

Exercise 

7 

1.   2.544. 

6.    .65959. 

11.    -  . 

4167.            16.    -f. 

21.   25. 

2.    1.2445. 

7.    -29.78. 

12.    .29414.               17.    -3. 

22.    /,. 

3.    2.495. 

8.    5.9837. 

13.   3. 

18.    -4. 

23.   32. 

4.    -.053474. 

9.    -  .46187. 

14.    5. 

19.    2. 

24.    17.677. 

5.    1.465. 

10.    .64509. 

15.    -2.                   20.    81. 

11.894. 

25.    5%. 

Exercise 

8 

1.               sin  B  = 

b                    b 

sec  5=^-, 

cos5=7»     cot  5= 

«       csc5=£. 

c'                   a' 

a 

c 

0                             0 

2.               sin  A  = 

f,      tan  ^4  =  |, 

sec  A  =  |, 

cos  .4  =  f,      cotJ.= 

|,      csc^4=f. 

3.               sin  A  = 

£,       tan^l  =  t, 

sec^l=f, 

cos  ,4=f,      cot  .4= 

|,      csc^4.=f. 

4.               sin  A= 

T8y,     tan  A  =  -jTf  , 

sec  A  =  \l, 

cosvl  =  !f,    cotJ.= 

J/,     csc^l=J^. 

5.               sin  ,4  = 

if,     tan  .4=^, 

sec^=V, 

cos  A  =  j5^,    cot  A  = 

r\,       080^=^1. 

6.               sin  .4  = 

||,     tan4=ff, 

secvl  =  -|§, 

cos^4  =  f|,    cotA  = 

|f,       CSC^l  =  ^f. 

7.               sin  ^4  = 

¥9T,    tan  A  =  £%, 

sec  A—fa 

cosA=±$,    cotA  = 

-4^)-,    csc  ^4.=^-. 

8.               sin  A  = 

Mf,  tan^4=ii§, 

secA  =  \%% 

,  cos.4=^!§,  cot  .4  = 

iff,    CSCu4  =  Hf. 

9.       III.  sin  5=|,       tan  5=f,       sec5=f, 

cos  5=f,      cot  5  = 

|,      csc5=f. 

IV.  sin  5= 

if,     tan  5=^, 

sec5=Y-» 

cos  5  =  yy}    cot  5  = 

r85,     csc5=4;|. 

V.  sin  5  = 

T53,     tan5=T\, 

sec5=}|, 

cos  5  =  4~?  ,     cot  5  = 

-1/,     csc5=1-53- 

VI.  sin  5  = 

|f,     tan  5=  f$, 

sec  5=  ft, 

cos  5=  If,     cot  5= 

ff,     esc  5=  ff. 

VII.  sin  5= 

!£,     tan  5=^°-, 

sec5=-V, 

cos5=¥\,    cot  5= 

3%,     csc  5=|£. 

VIII.  sin  5= 

if  a,  tan  5  -fff, 

sec  5=  if  \ 

;,  cos  5=  1  £f,  cot  5= 

i^f,  csc  5=  iff. 

10.    (1) 


(2)  1.       (3)  1.        (4) 


(5)  1.        (6)  1.        (7)  0.        (8)  1 


ANSWERS 


22.    AD  =  218.4. 


sin  A  = 


p*  +  q2 


28. 


30. 


p  +  q 

9 


Cl>  =  358.7. 
23.   .854. 

cos  A  = 


DB  =  181.3. 
24.    56.75. 


5 

=  283.86. 


p  +  q 
smA  =    \m«^  secA=^  +  nl 


tan  A  = 


2  mn 
m2-TO2< 


31. 


32. 
33. 

34. 
35. 


sin  *=™=-5, 
w»  + 

cos  5  = 

TO  +  71 

sin  .B=fff ,    tan  5=  ^,    sec  B=-\% 
sin  -<4=f  \/5,  tan  ^4=2,        sec^l=V5, 


=-,    tan  ^1= 


W  —  TO 


=|5,  cot  -A=£, 


cscJB=||f. 

CSC  4=^? 

2  ' 


36. 


6 


37. 
38. 
41. 

1. 

2. 
3. 


6  7  7 

sin  ^4 = ||,      tan  A = -^ ,      sec  A = -^ ,      cos  A  =  T\,      cot  vl  =  & ,     esc  ^4 = if. 

1.62.  42.   f,  f. 

Exercise  9 

cos  30°.  4.    tan  34°  24'.  7.    cos  1'. 

sin  75°.  5.    sec  68°  35' 30".        8.    sin  88°  42'. 

cot  24°  36'.  6.    esc  5°  44'.  9.    V3. 

Exercise  10 

sin  A = if,          sec  A = J^, 
sin  J.=i*, 


10.  4. 

11.  *• 

y 

12.  p. 


esc  A  =  Vm2  +  1. 


6.  sin  A  = 

7.  tan  A 

8.  sin  A 

9.  sin  A 

10.  tan  .4 

11.  sin  A  — 

12.  tan  x  = 


^  tan  A 
5 

:  0,  sec  A 

1,  tan  A 

0,  sec  A 

co ,  sec  A 
tan^l 


V5 

IT1 

cosA  =  1, 

sec  A  =  oo  , 
cos  -4  =  1, 
cos  A  =  0, 
=  0, 

1 


cos  A  — 


2V5 


cot  A  —  co  , 
cot  .4  =  0, 
cot  .4  =  co, 
cot  A  =  0, 
cot  A  =  0, 


cot  A  =  2. 

csc  .4  =  oo. 
csc  .4=1. 
csc  A  =  co. 
csc -4=1. 
csc  A  =  1 . 


cot  x  = 


13.  sin  A 

14.  sin  A 

15.  sin  .4 


16.    sin  A  = 


17.  tan  A 

18.  sin^l 


19.     sin  A  = 


2   '     ""  2   ' 

21.     sin  x  =  0,  tan  x  =  0,  sec  x  =  1 ,  cot  x  =  co ,  csc  x  =  oo . 

tan  A  =  4^,          sec  ^1  =  ^-,         cos  A  =  ?9T,          cot  A  =  ?9,j, 

ni2  +  ri2 


csc  J.  = 


J™*,.      cos  .4  =  ^-^,      cot^  = 

?>i2  +  n2 
2  mn 


2  mn 


sec  J.  = 


tan  A  =  V2  —  1, 
=  V4  +  2V2. 

cos  ^4  =  0,  cot  .4  =  0,  csc  .4=1. 


26.    sin  22  1°  =  £  V2  -  V2, 


cos 


_  V2  +  V2 


ANSWERS  7 

29.    sin  A  =  Vl  -  7f2,  ,  ,  , 


tan  15°  =  2  -V3,     cos  15°  = 


30 

sec  15°  =  2  V2  -  V3,     esc  15°  =  2  V2  +  V3. 


31.    cos  ^1  =  Vl  -  sin2 .4,    tan  ^  = 


32.    sin  A  =     l-  cos2  ^4,    tan  ^  =  Vl  -  cos2  A 


sec  A  =  --  ,   esc  A  = 

cos  A  Vl  -  cos2  A 


33.    sin  A=-          ^__,   cos.4=  —  -,cotA  = 

Vl  -|-  tan2  A  Vl  +  tan2  A     , 


2^,    esc  A  = 


tan^l 


34.    tan^4  =  —  — ,   esc  A  =  Vl  +  cot2  ^4,   sin  A  = 


Vl  +  cot2  A  cot  -A 

i  . . 

,   tan  A  =  Vsec2  A  —  1,   cot^l=  — 

Vsec2  A  -  1 


Vsec2  A 
36. 


. 
esc  A  Vcsc2  ^i  -  1 

sec  A  =  -  gg^   —  ,   cot  ^1  =  Vcsc2  ^-  1. 
Vcsc2  ^1  —  1 


37.    cos  A  =  1  —  vers  A,  sec  .4  = 


1-versJL' 

1  —  vers  A 


1  —  versJ.  v  2  vers  .4—  vers2  A 


sin  ^4  =  V2vers  A  —  vers2  ^4,   esc  A  = 


V2  vers  J.  -  vers2  A 

38-    1-  «•  A-  «• 


39.  rf^ViSTS.  43.  . 

40.  W3.  44.    4V42. 

48.  2  sin2  x  +  sin  x  =  1. 

41.  38?V39.  45.    1-  cos2  ,4  +  cos  A 

49.  tan2x—  2tanx  =  1. 


ANSWERS 


Exercise  12 


13. 

2i- 

17. 

-1-V2. 

22.   iV6.                    36. 

150;  259.8. 

14. 

iV3(&  +  c). 

18. 

-«i- 

23 

.   5. 

38. 

961.  3+. 

15. 

2+V2. 

20. 

£(v"2—  1). 

35 

.  86.6. 

39. 

165. 

16. 

1-2V3. 

21, 

*- 

Exercise 

13 

1. 

60°.        4. 

60°. 

7. 

45*.        10.   60°. 

13.  60°. 

16.   30°.         19.   60° 

2. 

60°.         5. 

0°. 

8. 

45°7/      11.  45°. 

14.   30°. 

17.  45°.        20.   90°. 

3. 

30°.        6. 

45°. 

9. 

3Q?/      12.  30°, 

90°. 

15.  45°. 

18.   45°.         21.   0°. 

22.   27°  13'  12".              26 

.   22£°. 

28.    18°. 

33.   30°. 

23. 

15°. 

27 

90° 

2 

9.  45°. 

34.   60°. 

24. 

10°. 

_1_   1 

30.  38°  50'. 

35.  30°. 

25. 

60°. 

Exercise 

14 

1. 

9.64647  - 

10. 

9. 

8.95017  -  10. 

19. 

6.1493. 

26. 

9.9523  -  10. 

2. 

9.98997  - 

10. 

10. 

9.97991  -  10. 

20. 

14.991. 

27. 

0.3076. 

3. 

9.86603  - 

10. 

11. 

0.11532. 

21. 

9.4214  —  10. 

28. 

0.6489. 

4. 

9.38699- 

10. 

12. 

9.99194  -  10. 

22. 

9.8297  -  10. 

29. 

9.8832  -  10. 

5. 

0.15908. 

13. 

1.24820. 

23. 

0.  175t). 

30. 

0.2522. 

6. 

9.43707  - 

10. 

14. 

8.91931  -  10. 

24. 

0.7033. 

31. 

0.6413. 

7. 

8.73767  - 

10. 

15. 

9.84324  -  10. 

25. 

9.6622  -  10. 

32. 

15.24. 

8. 

9.86126  - 

10. 

16. 

9.74610  -  10. 

Exercise 

15 

1. 

23°  15'. 

8. 

85°  5'  15". 

15. 

28.7°. 

21. 

61.07°. 

2. 

28°  40'. 

9. 

65°  10'  20". 

16. 

'18.5°. 

22. 

0.541°. 

3. 

35°  43'. 

10. 

5°  20'  29". 

17. 

56.26°. 

23. 

88.465°. 

4. 

40°  23'. 

11. 

4°0'47". 

18. 

70.14°. 

24. 

65.67°. 

5. 

66°  15'  24". 

12. 

85°  59'  13". 

19. 

64.43°. 

25. 

78.14°. 

6. 

70°  16'  21". 

13. 

26.5°. 

20. 

46.11°. 

26. 

14.47°. 

7. 

70°  0'  26". 

14. 

50.2°. 

Exercise 

16 

1. 

8.21421  - 

10. 

14. 

0°  4'  31". 

27. 

8.1238  -  10. 

40. 

4.662°. 

2. 

8.34812  - 

10. 

15. 

0°  2'  39". 

28. 

8.1070  -  10. 

41. 

84.35°. 

3. 

8.49128  - 

10. 

16. 

89°  45'  6". 

29. 

8.2701  -  10. 

42. 

8.3638  —  10. 

4. 

1.72220. 

17. 

42°  5'  26". 

30. 

1.6657. 

43. 

1.6362. 

5. 

1.64078. 

18. 

82°52'1". 

31. 

1.8744. 

44. 

89.266°. 

6. 

8.18538- 

10. 

19. 

83°  24'  25". 

32. 

8.3446  -  10. 

45. 

.613°. 

7. 

8.28456  - 

10. 

20. 

0°17'7.3". 

33. 

7.9686  -  10. 

46. 

89.285°. 

8. 

8.47866- 

10. 

21. 

0°  17'  7.1". 

34. 

89.266°. 

<47. 

.624°. 

9. 

0°  26'  10". 

22. 

89°  54'  15". 

35. 

1.036°. 

48. 

1.6375. 

10. 

88°  53'  6". 

23. 

8.245. 

36. 

89.216°. 

49. 

2.792. 

11. 

0°  42'  53". 

24. 

.1504. 

37. 

.634°. 

50. 

112.82. 

12. 

89°  32'  27". 

25. 

1.6687. 

38. 

89.553°. 

51. 

.7348. 

13. 

89°  57  •'. 

26. 

8.3353  -  10. 

39. 

.507°. 

52. 

.026694. 

ANSWERS 

9 

Exercise  17 

1.  Sine  4  =  T8r.          Cosine  4  =  ^f  .          Cotangent  4  = 

J/«          Secant  4  =  {|. 

Cosecant  4  =  ^.        6  =  30. 

c  =  34. 

2.      —   5T6^. 

8.  cot  37°  >  tan  37°. 

22.   1. 

5.  sin  49°  >  cos  49°. 

19.   x  =  45°. 

23.   fVS—  £\/2  —  f. 

6.   4<45°. 

20.   ic  =  60°. 

0.     11-3V3 

7.   4>60°. 

21.   x  =  45°. 

2 

25.  cot  4  =  ^,  esc  4  =  ^ 

26.    *?.         27.    .3056. 

28.   300.        29.  270.12 

r 

- 

Exercise  18 

4.   5  =  62°. 

7.  B  =  61°  43'. 

10.   5  =  51°  43'  36". 

a  =  6.3804. 

a  =11.  448. 

a  =  2.2478. 

c  =  13.591. 

6  =  21.276. 

b  =  2.849.  . 

5.   5  =  12°. 

8.  4  =  35°  17'. 

11.  4  =  17°  43'  18". 

a  =  26:15. 

a  =  648.67. 

6  =  70.985. 

b  =  5.5585. 

6  =  916.7. 

c  =  74.5217. 

6.   5  =  43°  42'. 

9.   4  =  52°  41'. 

12.    .23661. 

a  =  50.78. 

a  =  385.436. 

13.    .282726. 

c  =  70.24. 

c  =  484.644. 

14.    B  =  26°  31'  20". 

15.   4  =  2°  43'  30". 

16.   5  =  38°  50'  54". 

b  =  127.976. 

a  =  13.85129. 

a  =  .153254. 

c  =  286.  5875. 

b  =  .674616. 

b  =  .12343. 

17.    B  =  63°  41  '24". 

18.    .96565. 

6  =  256.406. 

19.    164.93. 

c  =  286.033. 

20.    1416.13. 

21.    1614.26  yd.  =  depth  of 

canon.    5521.125  yd.  =  distance  of  river. 

24.    5=57.4°. 

30.    5  =  68.68°.                     39 

'.    352.1. 

a  =  11.5125. 

b  =  41.65.                       41 

.  5=60°. 

c  =  21.37. 

c  =  44.71. 

a  =  £  V3  =  4.0425. 

25.    5  =  34°. 

31.   4  =  23.73°. 

c  =  i/  V3  =  8.083. 

a  =  2.22. 

a  =  .003824.                   4g 

.   a  =  b  =  6V2  =  8.484. 

b  =  1.4976. 

c  =  .009504. 

26.    4  =  51.8°. 

32.    .3907. 

c  —  ^\/3  =  28.86. 

a  =  .604. 

33.    .11388.                             44 

b  =  iAJLQ-v/3  =  577  4. 

b  =  .4753. 

34.    50.933. 

27.    4  =  7.5°. 

35.    5  =  1.83°.                        45 

&  1.2^0^3-1154  8* 

b  =  95.42. 

a  -  13.125. 

c  =  i.^LQ-v/3  =  2309.5. 

c  =  96.225. 

b  =  .4194.                       46 

.    a  =  600  V3  =  1039.25. 

28.    5  =  62.33°.  , 

36.    4  =  47.84°. 

5  _  goo. 

a  =  77.43. 
&  =  52.33.  ? 
29    4  =  13.75°. 

b  =  .4757.                       4- 
c  =  .7086. 
37.   129.15. 

.    a  =200. 

c  =  200V2  =  282.8. 

6  =  3.7845. 

H 

.     €L  —  J-lJ  Ct» 

c  =  3.89583. 

38.    1.081. 

&  =  10dV3=  17.32  d. 

10 


ANSWERS 


49.    Same  as  the  respective  answers  for  numbers  6  and  7  in  this  exercise. 
51.    Z>5  =  50.     BC  =  25.     DC  =  \5-VS  =  21.65 x. 

Exercise  19 


1.    A  =  35°  33'  27".              16. 

B  =  17°  56'  5".                31.    50.43. 

6  =  14.969. 

b  =  8.6188. 

32.    A  =  18.96°. 

2.    A  =  33°  18'  3".                17. 

13°  7'  18". 

a-  50.91. 

6  =  31.147.                       18. 

Z  =  67°  22'  48",             33.    B  =  7.812°. 

3.    A  =  42°  24'  43". 

.-.  7'  12"  too  small.                   b  =  117.166. 

b  =  29.2557.                    21. 

A  =  41.49°. 

34.    57.26°. 

4.    A  =  39°  48'  20". 

b  =  17.755. 

35.   26.77°. 

c  =  7.81016.                    22. 

^  =  45.17°. 

37.    ^4  =  #  =  45°. 

5.    A  =  49°  44'  5". 

a  =  .39855. 

c  =  13V2  =  18.384. 

b  =  .579587.                    23. 

A  =  50.66°. 

38.    ,4  =  30°. 

6.    .4  =  49°. 

c  =  43.04. 

b  =  9V3  =  15.888. 

a  =  16.3608.                    24. 

A  =  32.02°. 

39.    JB  =  30°. 

7.    A  =  52°  12'  25". 

c  =  9.432". 

a=100V3  =  173.2. 

c  =  .079471.                    25. 

A  =  46.31°. 

40.    J5  =  30°. 

8.   .4  =  43°  52'. 

a  =  7.015. 

c  =  2. 

b  =  .184875.                    26. 

^  =  48.43°. 

41.   ^  =  60°. 

9.    53°  7'  48". 

c  =  .19107. 

6  =  3. 

10.    21°  53'  58".                       27. 

^1  =  40.67°. 

42.   ^4  =  45°. 

11.    42°  24'  39". 

a  =  86.64. 

6  =  1. 

12.    c  =  8.48.                            28. 

A  =  40.95°. 

43.    .4=60°. 

13.    25°  48'  40". 

6  =  .0839. 

6  =  50. 

14.    B  =  16°  11'  7".                29. 

A  =  52.33°. 

44.   ^1  =  30°. 

b  =  32.702. 

c  =  2987.33. 

a  =  6. 

15.    A  =  8°  31'  31".                30. 

A  =  43.44°. 

c  =  12. 

a  =  53.666. 

Exercise  20 

1.              Leg  =  120. 

8. 

Base  Z  =  46°  16'  41". 

Vertex  Z  =  60°. 

Vertex  Z  =  87°  26'  38". 

2.            Base  =  353.87. 

Leg  =  6690.  16. 

3.            Base  =  9.6837. 

9. 

r  =  8.2583. 

Vertex  Z  =  67°  24'. 

R  =  10.208. 

4.              Leg  =  50.699. 

Perimeter  =  60. 

Base  =  79.578. 

Area  =  247.75. 

Vertex  Z  =  103°  24'  20". 

10. 

r=  1.5388. 

5.     Vertex  Z  =  69°  23'  12". 

R=  1.618. 

Leg  =  927.84. 

Perimeter  =  10. 

Base  =  1056.225. 

Area  =  7.694. 

6.              Leg  =  8.8204. 

11. 

Side  =  8.282. 

Base  Z  =  62°  10'. 

r  =  15.455. 

Vertex  Z  =  55°  40". 

Area  =  768. 

7.         Base  Z  =  33°  21'  30". 

12. 

Side  =  9.  112. 

Leg  =  .075978. 

r  =  17. 

Area  =  929.24. 

ANSWERS 


11 


13. 


14. 


15. 


16. 


Side  =  8.6524. 
r  =  5.9546. 
Perimeter  =  43.262. 
Area  =  128.8. 
Perimeter  =  4.70498. 
Area  =  1.6417. 
h  =  IsmD. 
ra  =  2  1  cos  D. 
C  =  180°  -  2  D. 


17. 


18. 


19. 


sin  D  =  " . 


cos  -  C  =  -  • 
2          I 


m  = 

C  =  180°  -  2  D. 

h  =  i  m  tan  Z>. 


Z  =  -i 

D  =  90°  -  A  C. 


fc  =  - 

2 


2 


V 

*/ 

I 

—      esc 

C. 

m 

2..      2 

tan  ^  ( 

'  ~2~7T 

20. 

Base 

=  3.889. 

BaseZ 

=  42°  15' 

34". 

Vertex  Z 

=  95°  28' 

52". 

21. 

12.7001. 

34. 

Base 

=  .0588. 

41. 

Side 

— 

9.318. 

22. 

95.94. 

Leg 

=  .  12027. 

r 

5= 

17.387. 

23. 

15.1848. 

35. 

BaseZ 

=  54.275°. 

Area 

= 

972. 

26. 

8.1183. 

Leg 

=  26.77. 

42. 

22.025. 

27. 

48.2055. 

36. 

Base 

=  .8462. 

43. 

111.4. 

28. 

Base 

=  61.86. 

BaseZ 

=  14.15°. 

44. 

Altitude 

— 

2/-\/3 

Vertex  Z 

=  114.8°. 

37. 

r 

=  16.9. 

— 

14.435. 

29. 

Leg 

=  2081.5. 

Area 

=  946.5. 

BaseZ 

E= 

30°. 

Vertex  Z 

=  45.2°. 

38. 

Perimeter 

=  143.166. 

45. 

BaseZ 

3= 

30°. 

30. 

Leg 

=  34.47. 

39. 

Side 

=  1.0878. 

Base 

= 

50  V3 

Base 

=  59.026. 

r 

=  1.6737. 

= 

86.6. 

BaseZ 

=  31.14°. 

40. 

Side 

=  20.22. 

46. 

2T-  ^3 

— 

11.547 

31. 

BaseZ 

=  52.86°. 

r 

=  21. 

3= 

leg  =  base. 

Leg 

=  .61014. 

E 

=  23.3. 

47. 

BaseZ 

= 

45°. 

32. 

BaseZ 

=  61.1°. 

Area 

=  1486.34. 

Vertex  Z 

= 

90°. 

Base 

=  124.4. 

Perimeter 

=  141.54. 

Altitude 

— 

6. 

33. 

Base 

=  114.2. 

48. 

120°. 

Vertex  Z 

=  114.54°. 

49. 

7.07. 

Exercise  21 


1.  12560.57. 

2.  5911.7. 

9.     b  =  3.416. 
c  =  4.2881. 

A  =  36°  11'  53". 

10.  a  =  2.67815. 
b  =  5.41875. 
c  =  6.0445. 

11.  a  =  13.1945. 
b  =  8.4405. 

A  =  57°  23'  36". 

12.  42.847. 


3. 

172.756.                       5. 

3122. 

7.    .19936. 

4. 

545.44.                         6. 

21519.5. 

8.    202281.818. 

13.    .088996. 

19. 

I  =  7.1773. 

14.    .0287326. 

• 

c  =  12.299. 

15.   244.79. 

h  =  3.7011. 

16.    300.61. 

20. 

.7723. 

17.    h  =  5.2496. 

21. 

9.58675. 

I  =  6.1403. 

22. 

1.5458. 

A  =  58°  45'  17". 

23. 

.8874. 

18.      1  =  1.5086. 

24. 

fi  =  3.22046. 

c  =  2.6811. 

c  =  2.2029. 

h  =  .69175. 

r  =  3.0263. 

12 

ANSWERS 

25.    Perimeter  =  21.265. 

42.    151.4. 

54.    72  =  18.34. 

26.    p  =  23.181. 

43.    80.8. 

c  =  10.3332. 

R  =  3.9448. 

44.    .2084. 

r  =  17.6. 

28.    938. 

45.    h  =  8.828. 

55.   22  =  4.031. 

29.   47577. 

A  =  22.03°. 

c  =  2.7575. 

30.    882. 

I  =  23.54. 

r  =  3.788. 

31.    .01618. 

46.    I  =  1.2351. 

56.    101.36. 

32.    31.47. 

ft  =  .7478. 

57.    2886.8  =  ift-jp  V3. 

33.    137.33. 

c  =  1.9656. 

58.    180000  \/3  =  301760. 

34.    6000000. 

47.    I  =  54.51. 

59.    298.78. 

35.    .00003529. 

c  =  91.06. 

60.   4050  v/3  =  7014.6. 

36.     a  =  8.283. 

h  =  30.04. 

61.   3200  V3  =5542.4. 

A  =  52.44°. 

48.    c  =  .8598. 

62.  800. 

c  =  10.45. 

h  =  .2384. 

63.    2000000^/3  =  3464000. 

37.     c  =  77.22. 

.4=29°. 

64.   7200. 

a  =  68.9. 

49.    58.75. 

65.    2500  V3  =  4330. 

6  =  34.84. 

50.   .8308. 

38.    Impossible. 

51.   36950. 

66.   mono  v/3  =  5773.3. 

39.    .13833. 

52.   15.172. 

67.    400  V3  =  692^. 

40.    149.07. 

53.    H  =  2.262. 

68.   80,000. 

41.   4813.3. 

c  =  1.9624. 

r  =  2.038. 

Exercise   22 

In  this  exercise,  where  two  answers  are  given  to  an  example,  the  first  is  the  result 

obtained  by  use  of  five-place 

log  tables,  and  the  second 

answer  is  the  result  obtained 

by  use  of  four-place  tables. 

1.    389.7  =  Ht. 

9.   695.414. 

19.    23.013. 

2.    474.788. 

695.35. 

23.012. 

474.8. 

10.    17°  31'  7". 

20.    5246.25. 

3.    114.1. 

17.52°. 

5246.6. 

4.    10°  33'  25". 

11.    82.056. 

21.   43.3  =  ht.  of  tree. 

10.56°. 

82.06. 

25     =  width  of  river. 

5.    491.511. 

12.   287.25. 

22.   KR  =  12. 

491.44. 

287.47. 

HP  =  6  V3~=  10.392. 

6.   Base  =  76.  79. 

13.   231.7. 

It  $  =6^/6  =  14.694. 

Base  =  76.8. 

231.68. 

/ST=  12V3  =  20.784. 

Alt.    =49.6955. 

14.    1534.96. 

SF  =  24. 

Alt.   =49.7. 

1535. 

TF  =  12. 

Area  =  1908.5. 

16.   Ht.  of  hill  1673.038. 

23.    13.071. 

Area  =  1908.08. 

Ht.  of  hill  1673.67. 

13.053. 

7.   37°  58'  46". 

Dis.  of  ship  621  5.  143. 

24.    71.264. 

37.975°. 

Dis.  of  ship  62  15.  7. 

71.28. 

8.    Distance  of  ladder 

17.   KR  =  12  V3  =  20.784 

.    25.    616.771. 

from  house  =  12.588. 

KA  =  24. 

616.5. 

12.58. 

KT  =  6  V3  =  10.392. 

26.   45°0'37". 

Z.  ladder  makes  with 

HT  =  18. 

45°. 

house  =  30°  14'  8" 

FT  =  18  V3  =31.176 

50.6375. 

=  30.22°. 

2tF  =  36. 

50.62. 

ANSWERS 


13 


27. 


AB  =  sin  y. 
OB  =  cos  y. 
BO  —  sin  x  cos  y. 
OC  =  cos  x  cos  y. 


29de  =  26.0.9 

A  =  108° 14' 40" 
108.26°. 
71° 45' 20". 
71.74°. 


Exercise  23 

1.  2.  3.    3.  5.   4.  7.    4.          9.    3.          11.    1.          13. 

2.  2.  4.    4.  6.    1.  8.    3.         10.    3.          12.    2.  14. 

16.    (1)  Same  as  the  signs  of  the  functions  in  the  second  quadrant. 
(3)  Same  as  the  signs  of  the  functions  in  the  third  quadrant. 
(5)  Same  as  the  signs  of  the  functions  in  the  fourth  quadrant. 


15.  4. 


17. 


21. 
27. 


34. 


385°.  18. 

745°. 

-  335°. 

-  695°. 

65°.  22.    60°. 

Second. 

Thirf 

8.052  (by  use  of  five-place  tables). 

55.73. 


330°. 

19.    460°. 

690°.  . 

820°. 

-  390°. 

-  260°. 

-  750°. 

-  620°. 

23.    60°. 

24.    155°. 

29.    Second. 

30.    Third. 

20.  260°. 
620°. 

-  460°. 

-  820°. 
25.  40°. 

31.  Fourth. 

32.  Second. 

8.06  (by  use  of  four-place  tables). 


26.    53C 


1.  2. 

2.  oo. 


Exercise   24 

3.  0.  5.    4. 

4.  c2  -  a2  -f  4  ac.       6.    —  2  a. 


7.  0. 

8.  3m. 


1,  sin  390°  =1. 
cos  390°  = 
tan  390°  = 

sec  390°  =  f  V3. 

2.  cos  780°  =  £. 
tan  780°  =  V3. 
sin  780°  = 
cot  780°  = 

4. 


5. 


6. 


sm  =: 
cos  =  |. 
tan  =  V3. 
cot  =  £-V3. 
sin  =  i. 
cos  = 
tan  = 
cot  =  V3. 
sin  = 
cos  = 
tan  =  1. 
cot  =  1. 


7. 


10. 


11. 


Exercise  25 

sin  =  i. 
cos  =  \  V3. 
tan  =  £\/3. 
cot  =  VS. 


sm  = 

cos  = 

tan  = 

cot  = 

sin  — 

cos  = 

tan  — 

cot  = 

sinx  = 

tanx  = 

cotx  = 

secx  = 

CSC  X  = 

sin  x  = 
cos  x  = 
cotx  = 


v§. 

1V3. 

JV2. 

iV2. 

1. 

1. 

±1- 

Tf 


±1- 
±if. 

T    ,V- 


12. 


13. 


14. 


secx  — 
cscx  = 

COSX  =: 

tan  x  — 

secx  = 
cotx  = 
cscx  — 


sin  x  =  — 


_   13 

x/5 
5 


COS  X  =  — 

tan  x  =  \. 
cot  x  =  2. 


•5 


sec  x  =  — 


v 


-    -A/5. 


sm  x  = 


Vm2-l 


cos  x  = 

m 


14 


ANSWERS 


cscx  = 


sec  x  =  VIO. 

Vio 

3 

V35 
6 


CSC  X  =  — 


sin  x  =  — 


18.  sin  y  =  —  ^V5. 
csc  y  =  —  |  VS. 

19.  sin  x  =  —  -i. 

V3 


COS  X  — 


2 


Vw2-l 


15.        sin  x  =  — 


10 


cosx  = 


VlO 
10 

=  -«. 

cot  x  =  —  4. 


1.  -J.    2.  J.    3. 

10.  --^+-5. 

11.  - 

12.  sin  38. 

13.  -tan  17°. 

14.  sin  40°. 

15.  —sec  5°. 

16.  tan  5°. 

34.  a  cos  x  -f  b  sin  x 


cos  x  =  —  I . 

tan  x  =  V35. 
sec  x  =  —  6. 

6V|5 
35 


tan  x  =  — 


CSC  X  =  — 


cotx  = 


V35 


cot  x  =  —  VS. 

V3 
3 

secx  =  2^§. 
csc  x  =  —  2. 


21     - 


VS. 


6.  - 


1. 

10.  sin  : 

COS  : 

tan: 

COt  : 

sec: 
csc 

11.  sin 
cos 
tan 
cot 
csc 
sec 

12.  sin; 
cos 
tan 
cot 
sec 
csc 


2.  VS.    3. 

:  —  COS  29°. 

:  -  sin  29°. 

:  COt  29°. 

:  tan  29°. 

:  -  CSC  29°. 

i -sec  29°. 
i  -  cos  9°. 
i  sin  9°. 

:  —  COt  9°. 

=  -  tan  9°. 
=  —  sec  9°.' 
=  csc  9°. 
-  sin  15°. 
=  —  cos  15°. 
=  -  tan  15°. 
=  -  cot  15°. 
-.  —  sec  15°. 
=  csc  15°. 


Exercise  26 

4.  -VS.    5.  -V2. 

17.  -tan  45°. 

18.  -  sin  20°. 

19.  -sin  27°. 

20.  -cot  25°. 

21.  sec  30°. 

22.  -sin  27°. 

23.  cot  22°. 

24.  -cos  10°  16'. 

—  c  tan  x. 

36     —  (a  +  &)  cos  x  —  (a  —  6) 

Exercise  27 

_i.    4.  _V3.    5.  -VS.    6.  0. 

13.  sin  =  —  sin  15°. 
cos  =  cos  15°. 
tan=  —  tan  15°. 
cot  =  —  cot  15°. 
sec  =  sec  15°. 
csc  =  —  csc  15°. 

14.  sin  =  cos  17°. 
cos  =  —  sin  17°. 
tan  =  - cot  17°. 
cot  =  -  tan  17°. 
sec  =  — csc  17°. 
csc  =  sec  17°. 

15.  sin  =  cos  10°. 
cos  =  sin  10°. 
tan  =  cot  10°. 
cot  =  tan  10°. 
sec  =  csc  10°. 
csc  =  sec  10°. 


-1. 

M- 

V/8.      8.    -f.      9.    -J. 

25. 

-  cot  30°  17'. 

26. 

-  sec  25°. 

27. 

sin  8°. 

28. 

-  tan  20°. 

29. 

-  cot  30°. 

32. 

9|. 

33. 

11  cosx. 

35. 

p  sin  x  cos  x. 

sinx. 

7. 

-2. 

8.  i  V§.    9.  -iV2. 

16. 

sin  =  sin  0°. 

cos  =  —  cos  0°. 

tan  =  tan  0°. 

cot  =  cot  0°. 

sec  =  —  sec  0°. 

csc  =  csc  0°. 

17. 

sin  =  sin  36°  43'. 

cos  =  -  cos  36°  43'. 

tan  =-  tan  36°  43'. 

cot  =  —  cot  36°  43'. 

sec  =  -  sec  36°  43'. 

csc  =  csc  36°  43'. 

18. 

sin  =  cos  37.  24°. 

cos  =  sin  37.24°. 

tan  =  cot  37.24°. 

cot  =  tan  37.24°. 

sec  =  csc  37.24°. 

csc  =  sec  37.  24°. 

ANSWERS 


15 


21. 
22. 
28. 
29. 

—  COS  X. 

—  COS  X. 

—  a  cos  x  + 

—  ?>l  COS  J[  - 

23.    —  sinx.                 25. 
24.    t&nx.                     26. 
b  sin  x  —  c  tan  x.                  30. 
-#  cot  .4  —  g  cot  A.              31. 

Exercise 

—  sec  x. 
—  sec  x. 
sin2  x  cos  x. 

—  COS  X. 

28 

27.   —  3  cos  x. 

1. 

30°, 

150°. 

5.   30°, 

150°. 

9. 

45°, 

225°. 

2. 

30°, 

150°, 

210°, 

330°. 

6.   60J, 

300°, 

180 

°.      10. 

60°, 

240°. 

3. 

45°, 

135°, 

225°, 

315°. 

7.    30°, 

150°. 

11. 

45C, 

225°. 

4. 

30°, 

150°, 

210°, 

330D. 

8.    45°, 

225°. 

12. 

45°, 

135°,    225°, 

315°. 

13. 

30°, 

150°, 

45°, 

225D, 

315°. 

15. 

30°, 

150°, 

210°, 

330°, 

14. 

60°, 

120°, 

240°, 

300°, 

60°, 

120°, 

240°, 

300°. 

45°, 

135°, 

225°, 

315°. 

16. 

30°, 

150°, 

210°, 

330°. 

17. 

30°, 

150°. 

18.   30°, 

150°. 

Where  two  answers  are  given,  the  first  answer  is  found  by  the  five-place  tables, 
the  second  answer  is  found  by  the  four-place  tables. 

19.  66.35  mi.  east.     66.34  mi.  east.     27.14  mi.  north. 

20.  39°  10' 25".     39.18°.  21.   760.316.     760.33. 
22.   Distance  of  the  spring  from  the  house  =  217.39.     217.4. 

Distance  of  the  spring  from  the  barn    =229.12.     229.16. 


1. 

2. 
3. 

8. 
14. 

sin(x  +  2/)  =  f| 
sin  (x  -  y)  =  f 

cos(x-*/)=£. 
sin  (x  +  45°)  = 

cos  (30°  x) 

Exercise  29 

;.                                            4.    (x  +  y)  =  co  . 
|.                                         5.  cot  (x  —  y)  =  0. 

1-                                           c     V6-V2    ' 

90°  =  1. 
90°  =0. 

1                                                         4, 
—  (sin  x  +  cos  x).           7.    2  +  V3. 

sin  (x  -  60°)  = 

V6  4-  V2       0 

2 
slnx  —  cosxV3 

2 
,/Q               in    V6-V2        tt     V2-V6       12.    sin 

4 
tan  ("45°  -4-  v^  — 

4                             4                     cos 
1+tany              15     cot/60o  -     N      \/3cot2y-4coty 

'1-tany                                                          3cot2y-l 

1  +  tany                      C°t(3°    '  y)             cot22/-3         ' 

Exercise  30 


1. 

cos  60°  =  $. 

2.     tan  60°  =  V3. 
3. 


9.  3  sin  x  —  4  sin3  x. 


10.   4  cos3  x  -  3  cos  x. 
ii     3  tan  x-  tan3  a;  t 
l  —  3  tan2  * 

13-  -  ¥• 
14.  -/?. 
21.  £cos4x  +  £  cos  2  a;  -f  f. 


16 


2.  sin  15°  =  i  V2—  V3  =  .2588. 
tan  15°  =  2  —  A/3  =  .2679. 
cos  15°  =  i  A/2  +  V3  =  .9659. 

3.  cot  221°  =  V2  +  1  =  2.4142. 

cos  221  =  i  V2_+V2  =  .9239. 
sin  221  -  i  V2  -  \/2  =  .3827. 

4.  sin45°  =  cos45°=:i-\/2  =  .7071. 
tan  45°  =  cot  45°  =  1. 

sec  45°  =  esc  45°  =  v/2  =  1.4142. 


ANSWERS 
Exercise  31 
6. 


cos  -  =  _  V2  +  2  a. 
2      2i 


cot^  =  - 
2      1-a 

tan  i  =  — 1_ 

2      l+o 


12.       cos.i=A/1+COB2^. 

\  9 


/I  —  cos  2  ^4 


1  —  cos  2  ^1 


13. 


14.    - 


16. 


13. 


A  =  79°  36'  40". 
A  =  79.726°. 


3  \/5  +  25 
21 


17.    2°  44' 40". 

2.744°. 
Exercise  32 

14.    sin  (60°  +  30°)  =  1. 


sin   J.  —  ^  = 


_  VT5-\/3 


sin  60°  +  sin  30°  =  v  d  +  f , 

2 


8 


15.     - 


sin  29.5°  cos  7.5C 
sin  27°  sin  11° 


16.  -. 

cos  6  A    • 

17.  sin  (.4 +  5)  sin  (.4 -J5). 

18.  3.44. 
.2136. 


cos  2  5  = 


1. 


esc  e  =  — 

cot  9  =  4-. 


Exercise  34 
5. 


tan  (180° -0)  =  - 


cos  15°  =  i  A/2  +  \/3. 
Csc  15°  =  2  A/2  +  V3. 
tan  15°  =  2  —  V3~. 


the   sign   depending    on  whether   \x  is 
taken  in  the  first  or  fourth  quadrants. 
In  like  manner : 


6.    (a) 


(0 


10 


10 


10 


ANSWERS 


17 


=  -V3. 


00 


sin  (?r  —  0)  =  sin  6. 
COS  (TT  —  0)  =  —  COS  0. 
tan  (TT  -  0)  =  -  tan  0. 
cot  (TT  —  0)  =  —  cot  0. 

_i»\= 


w 

11 

oiu    i   jt/  i  —  \^\jojiy. 

(0 

25  V3  -  48 

(c} 

cos  i  x     —  ~  I  —  ••  sin  x 

39 

en 

V5 

vv/ 

tan  (x—   —  j  =  —  cot  x. 

7.    (a) 

2 

=  1- 

cot  (x  J  =  —  tan  x. 

(c) 

o 

sin  (TT  +  x)  =  —  sin  x. 

(*) 

-I  VS. 

/  ^7\ 

COS  (TT  +  x)  =  —  COS  x. 

sin  I  x  —  —  \  =  —  cosx. 

(c?) 

tan  (TT  +  x)  =  tan  x. 
cot  (IT  +  x)=  cotx. 

8.    (a) 

cos  {  x  —  —  j  =  sin  x. 

V  w/ 

tan  [  x  —  -^  =  —  cotx. 

cot  1  x  —  —  j  =  —  tan  x. 

34.    -£. 

35.    -|. 

36.   f.                37.    -|V3-                38-    ~2&- 

39.   tan  0  = 

=  |.              41.    _i|. 

K0     3  —  4  cos  4  x  +  cos  8  x 

sin  0  : 

—  f 

128 

54.    r£¥(35  -  64  cos  2  x  +  32  sin2  2  x  cos  2  x  +  28  cos  4  x  +  cos  8  x). 

Exercise  35 

3. 

a  =  c  cos  ^. 

7.    (I) 

*            —  tan  (  A        4*V^  nr 

d  a  right  triangle 

—    Lclll   I  _/l           ^O      )    dl 

(II)  a  +  &=(a-6)(2  +  A/3) 

an  isosceles  triangle  with  the  angles  30°,  30°, 

120°. 

9. 

smB  =  -< 

a 

smA  =  ^ 

Exercise  36 

1.     c  =  9.1226.                       4.    A 

=  109°  19'. 

7.    A  =  99°  29'  12. 

C  =  41°7'.                               a 

=  4899.56. 

6  =  1.0943. 

b  =  13.288.                              b 

=  4106. 

c  =  .  488667. 

2.   A  =  134°  20'.                     5.    C 

=  69°  57'  36". 

8.    B  =  43°  18'  36". 

6  =  74.9916                             a 

=  .85442. 

6  =  1.3487. 

c  =  242.755                              6 

=  .81196. 

c  =  1.8286. 

3.    .4  =  57°  52'.                       6.    A 

=  29°  1'  2'. 

9.    C  =  68°  15'  30'  . 

a  =  1116.98.                            a 

=  56.541. 

a  =  .182095. 

c  =  1260.26.                            6 

=  90.164. 

6  =  .188745. 

18 


ANSWERS 


10. 


16. 


11. 


12. 
13. 


14. 


15. 


b  =  5.267  V2. 

=  7.4486. 
c=  2.6335  (V6+V2). 

=  11.175.  17. 

C  =  105°. 
C  =  75°. 
a  =  500(3V2  -V6).18. 

=  896.55. 
6  =  600(2  V3-  2). 


c  =  38.52. 
6  =  57.412. 
.4  =  79.9°. 
a  =  13283.34. 
c  =  13346.67. 
A  =  80°  46'. 
a  =  600.4. 
6  =  602. 
C  =  .75°. 

=  732.1.  19.        c  =  7.295. 

4.0954.     11.697.  6  =  14.83. 

b  =  17.08.  A  =  117.67°. 

c  =  15.097.  20.        b  =  .2592. 

(7=56.73°.  a  =  .2181. 

a  =  634.3.  (7=55.87°. 

6=632.89.  21.        a  =  186.25. 

^  =  81.32°.  c  =  32.5. 

c  =  1.022.  A  =  101.96°.  ! 

a  =  1.4815.  22.        c  =  4377. 

B  =  25.57°.  6  =  5641.43. 

A  =  55.69°. 

30.  Distance  of  balloon  from  first  point  =  2033  yd. 
Distance  of  balloon  from  second  point  =  2363  yd. 
Height  of  balloon  =  1739  yd. 


23.  a  =  20.343. 
c  =  28.66. 

5  =  27.77°. 

24.  a  =  838.83. 

6  =  595.1. 
C  =  56.6°. 

25.  6  =  c  =  a  =  100. 
B  =  C  =  A  =  60°. 

26.  (7  =  30°. 

a  =  200  V3  =  346.42. 
6  =  c  =  200. 

27.  (7  =  45°^ 

6  =  250(3  V2-  V6)  =448.3. 
c  =  250(2  V3- 2)  =  365.7. 

28.  5  =  30°. 

c  =  200V2  =  282.,8. 

a  =  100(  V6  -j-  V2)  =  386.4. 

29.  925.8. 


Exercise  37 


1. 


2. 


3. 


4. 


5. 


6. 


c  =  26.8675. 
B  =  39°  45'  17". 
A  =  72°  14'  43". 
a  =  385.43. 
B  =  74°  38'  19". 
C  =  37°  3'  41". 
O=  110°  22'  10". 
5  =  39°  25'  30". 
a  =  .1912. 
A  =  48°  42'  12". 
C  =  67°  42'  18". 
b  -  .0748566. 
C  =  34°  6'  36". 
B  =  22°  36'  54". 
a  =  4.70177. 
a  =  336.446. 
B  =  99°  55'  36". 
C  =  27°  58'  24". 


7. 

8.185 

5= 

C. 

13. 

7? 

= 

141.99°. 

8. 

C 

— 

109°  36'  5". 

A 

— 

25.89°. 

B 

— 

38°  5'  25". 

c 

= 

3.972. 

a 

= 

14.962. 

14. 

A 

— 

79.82°. 

9. 

C 

= 

6°  49'  41". 

C 

— 

21.56°. 

b 

= 

317.8. 

b 

— 

1712.3. 

A 

— 

4°  51'  41". 

15. 

a 

= 

7.93. 

10. 

A 

— 

49.06°. 

16. 

B 

— 

6.23°. 

c 

= 

208.1. 

C 

= 

4.97°. 

B 

— 

79.117°. 

a 

= 

5.906. 

11. 

a 

= 

.9418. 

17. 

c 

= 

102.425. 

B 



117.99°. 

A 

— 

65.83°. 

C 



33.85°. 

B 

— 

45.93°. 

12. 

A 

— 

32.24°. 

18. 

A 

= 

33.84°. 

C 



35.58°. 

B 

= 

102.98°. 

b 

— 

.6566. 

c 

= 

1474.67. 

19. 

b 

= 

10.7. 

Where  two  answers 
place  tables,  and  the 
tables. 


are  given,  the  first  answer  is  obtained  by  using  the  five- 
second  answer  is  obtained  by  the  use  of  the  four-place 


ANSWERS 


19 


20.  Distance  =  234.34  ft. 
Distance  =  234.32  ft. 

21.  4.36  mi. 

22.  Kesultant  =  14.989. 
Resultant  =  15.08. 

Z  with  OA  =  77°  11'  20". 
Z  with  OA  =  77.23°. 
Zwith  0.8  =  43°  31' 40". 
Zwith  05  =  43.49°. 


23. 


3.59. 

152.268. 

152.22. 

238.31. 

238.22. 


Exercise  38 


1. 

A 

= 

78°  5'  36". 

78.1°. 

B 

— 

58°  23'  28". 

58.38°. 

C 

= 

43°  30'  58". 

43.52°. 

2. 

A 

— 

44°  32'  4". 

44.53°. 

B 

= 

86°  25'.     86, 

41°. 

C 

= 

49°  2'  58". 

49.05°. 

3. 

A 

— 

26°  19'  54". 

26.33°. 

B 

= 

98°  18'  54". 

98.32°. 

C 

= 

55°  21'  14". 

55.36°. 

4. 

A 

= 

45°  11'  50". 

45.19°. 

B 

— 

101°  22'  18" 

.     101.38 

C 

= 

33°  25'  58". 

33.43°. 

5. 

A 

— 

43°  53'  14". 

43.88°. 

B  =#0°  3'  36". 

60.06°. 

C 

= 

76°  3'  18". 

76.06°. 

6. 

A 

= 

61°  53'  38". 

61.88°. 

B 

= 

72°  46'  4". 

72.78°. 

C 

= 

45°  20'  20". 

45.34°. 

7. 

A 

=  91°  48'.     91.80°. 

B 

— 

47°  36'  56". 

47.61°. 

C 

= 

40°  35'  10". 

40.59°. 

16. 

.53224.     .5323. 

8. 

A 

— 

37°  50' 

40". 

37.84°. 

B 

— 

127°  3'. 

127.05°. 

C 

=  15°  6'  22". 

15.11°. 

9. 

A 

= 

40°  38' 

22". 

40.64°. 

B 

— 

49°  21' 

56". 

49.36°. 

C 

= 

89°  59' 

46". 

90°. 

10. 

A 

= 

52°  20' 

30". 

52.34°. 

B 

=  107°  19'  12", 

,     107.32°. 

C 

= 

20°  20' 

26''. 

20.34°. 

11. 

A 

— 

13°  12' 

8". 

13.2°. 

B 

=  30°  2'  46". 

30.04°. 

C 

=  136°  45'  6". 

136.76°. 

12. 

A 

- 

46°  19' 

52". 

46.33°. 

B 

— 

31°  17' 

50". 

31.3°. 

C 

= 

102°  22 

'  18" 

.     102.37°. 

13. 

A 

— 

107°  55 

'  12. 

107.92°. 

B 

— 

35°  15' 

34". 

35.26°. 

C 

= 

36°  49' 

18". 

36.82°. 

14. 

104° 

28'  42" 

.     104.48°. 

15. 

16 

o  44,  6// 

16.736°. 

21. 


17.    .1188.  18.    14.8586.     14.86. 

Q  is  53°  7'  48"  (53.14°)  north  of  west  from  P. 
Q  is  38°  52'  48"  (38.88°)  north  of  west  from  R. 
P  is  due  west  of  11. 

P  is  36°  52'  12"  (36.86°)  east  of  south  from  Q. 
R  is  due  east  of  P. 

R  is  38°  52'  48"  (38.88°)  south  of  east  from  Q. 
When  R  is  northeast  from  P  : 
Q  is  8°  7'  48"  (8.14°)  north  of  west  from  P. 
Q  is  6°  7'  12"  (6.12°)  south  of  west  from  R. 
R  is  6°  7'  12"  (6.12°)  north  of  east  from  Q. 

P  is  southwest  from  R.    P  is  8°  7'  48"  (8.14°)  south  of  east  from  Q. 
28°  57'  17".     28.96°. 


20 


ANSWERS 


Exercise  39 


1. 

2. 
3. 
4. 
5. 
6. 
7.v 
8. 
9. 
10. 

One  solution.                    15.    A  =  32°  55'  57". 

A'  =  147°  4'  3". 
Two  solutions.                           ^  _  131o  33;  51/; 

One  solution.                            C'  =  17°  25'  45". 
c  =  1643.96. 
No  solution.                               ct  -  661.15. 

No  solution.                      jg.    A  =  43°  38'. 

One  solution.                             5  =  58°  3'  42". 
6  =  .32868. 

One  solution,  a  right  A. 
17.    A  =  90  . 

No  solution.                                c  =  25.64. 
Two  solutions.                  18.    B  =  28°  16'  25". 

B      32°  36'  33"                        C  =  20°  25'  11". 
b  =  .56045. 

22.    .4  =  25,22°. 
(7=49.51°. 
a  =  135.46. 

23.    .4  =  20.79°. 
B  =  132.99°. 
b  =  136.733. 

24.    A  =  16.25°. 
4'  =  163.75°. 
C  =  149.45°. 
C"  =  1.95°. 
c  =  36.63. 
c'  =  2.4518. 

25.    5  =  122.81°. 
B'  =  12.45°. 

C  =  109°  5'  27". 

C  =  34.81°. 

c  =  211.48. 

19.    A  =  103°  50'  22". 

C'  =  145.19°. 

11. 

B  =  40°  40'. 
B'  =  16°  44'. 

A1  =  13°  7'  8"  =  A. 
a  =  15.354. 
a'  =3.589. 

b  =  441.7. 
6'  =  113.2. 

0=78*2'. 

B  =  44°  38'  23". 

26.    A  =  70.78°. 

C"  =  101°  58', 

B'  =  135°  21'  37". 

C  =  45.91°. 

b  =  15.787. 

a  =  10.08. 

b'  =  6.9753 

20.    A  =  35.91°. 

A'  =  144.09°. 

27.    4  =  72.16°. 

12. 

B  =  42°  44'  23". 

C=  111.72°. 

A'  =  9.22°. 

A  =  33°  1'  49". 

C"  =  3.54°. 

B  =  58.53°. 

a  =  92.942. 

c  =  219.7. 

5'  =  121.47°. 

13. 

,4  =  18°  19'  43". 

c'  =  14.6. 

a  =  .19685. 

C  =  139°  17'  59". 

21.    5=55°. 

a'  =  .03313. 

c  =  1.3952. 

B1  =  10.26°. 

14. 

5  =  70°  47'. 

C  =  67.63°.  . 

B'  =  14°  35'. 

C"  =  112.37°. 

> 

C  =  61°  54'. 

&  =20.118. 

C'  =  118°  6'. 

6'  =  4.372°. 

b  =  128.465. 

b'  =  34.2515. 

* 

2 

8. 

f  129.1. 
Other  side  =mi25< 

29.   1010.58. 
1010.2. 

Other  diagonal  = 


{  173°  15'  8". 
Larger  angle  of  parallelogram  =  1  17g  26° 


Smaller  angle  of  parallelogram 


_  f  6°  44'  52". 

~  [6.74°. 


ANSWERS 

Exercise  40 

1.    106.79. 

106.8. 

4. 

14290.6. 
14290. 

2.    .30733. 
.30726. 

5. 

38983.64. 
38983.33. 

3.    125.229. 
125.225. 

11.    Area  of  parallelogram 
13.   600  V3=  1039.2. 

6.    113.55. 
7.    .054776. 
.0547875. 
=  cd  sin  A.         14. 

106.798. 
106.8. 

Exercise  41 

8. 

9. 
10. 


1056.66. 

1056.25. 

1283.5. 

42150. 

42130.77. 


In  this  exercise  when  two  answers  are  given'  to  an  example,  the  first  answer  is 
found  by  the  use  of  five-place  tables,  and  the  second  answer  is  found  by  four-place 
tables. 


4.   69.372.  5.    72.268. 

69.37.  72.27. 

6.    8968.5  ft.  above  the  Colorado  plain. 

8958  ft.  above  the  Colorado  plain. 

14144.5  ft.  above  sea  level. 

14134  ft.  above  sea  level. 


7. 

373.3.          11. 

Height  = 

97.083. 

8. 

69.98. 

Height  = 

97.08. 

9. 

136.9. 

Distance 

=  71.787. 

Distance 

=  71.78. 

10. 

1016.6. 

1016.8. 

10.274. 

6.61. 

13. 

16.83. 

14. 

Other  side 

=  43.43. 

15.  Height  =  42.93. 
Height  =  42.92  ft. 
Distance  =  104.63. 
Distance  =  104. 675  ft. 

16.  11.36.  18.    4.2818. 
5.573.  4.283. 

17.  .1189.  19.    1496.517. 

1496.66. 

20.    First  answer  =  4.4867  mi.,  4.488  mi. 
Second  answer  =  9.16  mi. 


21. 


Other  diagonal  = 


I  58.342. 
{  58.346. 


24. 


996.94. 

997.6. 

401.52. 

401.54. 

443.54. 

443.5. 

974.145. 

973.9. 


25.    220.7. 


26.    16.58. 


30. 


146°  52'  47". 
146.88°. 
33°  7'  13". 
33.     12°. 

Difference  of  latitude  =  difference  of  departure  =  247.5  mi. 

New  latitude  =-34°  23'  North. 

New  longitude  =  48°  9'  W. 

152.69ft.  31.    114.5ft. 

152.7  ft. 

85.854  ft. 

85.89  ft. 

38.566  ft.  =  distance  of  first  observer  from  the  rock. 

2008  =  resultant. 

72°  16'  1 

0  [  =  angle  the  resultant  makes  with  OX. 
4  2.27     I 


27.  6739m. 
6740  in. 

28.  9°  6'. 


=  distance  between  observers. 


22  ANSWERS 

35.   298.  39.   367.89  ft.  j  =gide  opposite  tower. 


37.  161.8ft.  90.04  ft.  and  |       the  other  two  sides 

38.  97°  2'  32  379.125ft.  respectively. 
97.06°  and                                                        379>1  ft. 

14°  57'  28" 

14.94°  respectively. 

40.  48  ft.  and  108  ft.  respectively. 

41.  40°  0'  16"  1  _        le  the  gl        makes  with  the  embankment. 
40°  J 

29.45  ft.  =  width  of  base. 

42.  161.3.  43.   22°  49'  46".  44.   85.27  mi. 

22.83°. 


Exercise  42 

ofto     T  2        T_3o°  3.  1°  =  .  01745  radian. 

=  6  '  e  -  16"  =  .00007757  radian. 

2'  16"  =  .0006545  radian. 
135°  =  ££•  |  =  45°.  5o  14,  =  .0913374  radian. 

4.         2  radians  =  114°  35'  30". 

3=  3.2  radians  =  183°  20'  48". 

.003  radian  =  0°  10'  18.8". 

90°  =  —  -  v  =  120° 

23  5.    Arc  21  in.  long  =  f  radian. 

7  _  4  v  Arc  7  in.  long  =  \  radian. 

210°  =  —  •  "-as  144°. 

6  5  6.    #  =  28. 

270°=  —  -  P-Z^iQS0.  7.    Radians  =  1.118. 

Angle  =  64°  3'  22.5". 

225°  =  *£.  1-~  =  252°.  8.      Angles  =  85°  ;  25°. 

=  1.47325  radians;  .43625 

72°  =  —  .  §-^  =  96°.  radian. 

5  15 

315°  =  ^. 
9.    Complement  of  '  =  -  ,  60°  ;     supplement  =  ^  ,  150°. 

Complement  of  -  =  £  ,  30°  ;     supplement  =  ^  ,  120°. 

3  6  o 

Complement  of  -  ,  45°  =  -  ,  45°  ;    supplement  =  —^  ,  135°. 

4  4 

Complement  of  £  =  7~  ,  70°  ;  supplement  =  -  T  ,  160°. 
9       18  " 

Complement  of  |f  =  ^  ,  40°  ;  supplement  =  1^  ,  130°. 


ANSWERS  23 

10.    sin£  =  ^.  cos  =  -V3.  sin^  =  iV2.       cos  =  -^V2. 

62  2  42  2 

tan  =  i  V3.       cot  =  V3.  tan  —  cot  =  —  1. 

sec  =  |  V3.       esc  =  2.  sec  =  — V2.      esc  =  VI. 

sin^  =  iV3.       cos  =  --  Sm^=-i          Cos=-lV3. 

3  2_  2  6  2 

tan  =  V3.          cot  =  £  V3.  tan  =  J-  V3.        cot  =  V3. 

sec  =  2.  esc  =  |  V3.  sec  =  —  |  \/3.    esc  =  —  2. 

sin  »  =  cos  *  =  1  V2.  8inZr  =  -  5  V^   COS  =  5  v/1 

442  2 

ten»^ooC?  =  I.  tan=cot  =  -l. 

4  4  sec  =  V2.          esc  =  —  V2. 

sec7'  =  csc^  =  V2.  11.    H  radians  =  68°  45'  18". 

4  4 

sin^  =  l.    cot-=0.  13.    B  =  4. 

^  =  143°  14'  22.5". 
cos-  =  0.     sec -=oo.  14.    a  =  12.5. 

A  =  14°  19'  261". 
tan- =  00.    csc-=  1.  15.     p  =  8. 

22  A  =  458°  22'. 

16.   p  =  .26175.  17.    p  =  .64565.          18.   4'  35".  20.  4'  20". 

a  =  10.9935.  It  =  154.89.         19.    69.102ft.  21.    1117mi. 

22.   437320  mi.  23.    35374500  mi.  24.  f  V2  -  6. 

Exercise  44 


2. 

7T 

2ir 

4*- 

STT^ 

5. 

7T 

57T 

3' 

2 

8 

3   * 

6' 

6  ' 

3. 

7T 

37T 

STT 

77T 

6. 

7T 

STT 

4' 

4  ' 

4 

4,  ' 

3' 

3  ' 

4 

IT 

27T 

4ir 

STT 

7. 

7T 

S^r       T^r       ll^r 

3' 

3  ' 

3 

3  ' 

6' 

o        o         Q 

7T 

37T 

57T 

77T         7T         2lT 

47T       STT 

. 

4' 

4  ' 

4 

4  '    3  '      3  ' 

3  '      3  ' 

9. 

7T 

3_7T 

7T 

5jr 

16. 

7T 

7?r 

2' 

2 

6' 

6 

6' 

6  ' 

10. 

?r, 

iz. 

17. 

* 

P-Z,  Lz,  HE,  o,  z. 

6' 

6 

6' 

Q           O            u                  o 

11. 

0°, 

IT. 

18. 

7T 

3fl-         7T         57T 

4' 

4  '    3'      3 

12. 

37T 

4 

7  7T 

4 

7T 

'    2 

3jr 
2 

19. 

o, 

7T         7T          5  7T 

3'    6'    IT' 

13. 

i- 

T1 

f' 

T'  T'  T' 

20. 

o, 

TT       TT       5?r      ^      3?r 
2  '    6  '      6  '   T'      2 

— ,       ,      7T.                                                                                   21         0        7T                      - 

*           '        Q  '  Q'  Q'  O 

O  o  o  o 

-          ^  7T          4  7T          5  7T                                                            go         ^T          7T          7T  2  TT  4  7T  5  7T 

oooo                                                         o        ^       o  o  o  3 


4 

ANSWERS 

Exercise  45 

1.    0  =  30°,  210°. 

4.   sc=50°.                               7.    rK  =  36.87°. 

x  =100,  -100. 

2/  =  40°.                                     2/  =  22.62°. 

2.    0  =  36.5°,  216.5°. 

5.   a;  =  1000.                             8.   z  =  1000, 

z  =  200,  -200. 

2/  =  2000.                                   0  =  72.5°. 

3.    0  =  58.51°,  301.49°. 

6.    x  =  60°.                              9.   x  =  acos^  +  6sinA 

x  =  500,  -500. 

y  =  45°.                                    y  =  b  cos  J.  —  a  sin  .4. 

Exercise  46 

4 

2.                   cos  (cot-1  f)  =  f  . 

-lA/S      60°    * 

3.                 tan  (sin"1  T5j  )  =  r5^. 

'  3* 

4.                 sec(tan-1T^)  =  H. 

sin-1  4—30°    T  . 

5             •        sin  (cot"1  rt^  —                • 

'  6 

sec-i-y/2     45°    *"• 

6                  cot(co-lfl5)-aV^-:¥2 

'  4 

csc-12V3-60°   -. 

7.                tan  (2  sin-1  1)  =  V3. 

3 

8.               sin(2tan-iA)  =  Hft. 

'  6  " 

9.               cos  (2  sec-1  V)  =  -  Mi- 

JQ                                                     gjjj     /  1     CQg-l     1  N    lV3 

COB-4=W»,J. 

^2                  SJ—  3 

3 

11.              cot  (  J  tan-1  Y)  =  ±  f  • 

sec-1  2  =  60°,  -. 

12.                 sin  (3  sin-1  i)  =  1. 

3 

13     -in  (-in-"      CO--1*)     2~V^, 

sin-1|V3  =  60°,  |. 

6 
14.   tan  (tan  -1  2  +  cof1  3)  =  7. 

/~                      T 

3 

30.    "±2nir,                   31.    -  ±  2  WTT, 
6                                         6 

tan"1  ^  V3  =  30°,  —  • 

^±2»r.                      7^±2^. 

3,       f±S~, 

35.       -±2w7r,                        38.       7r±2mr, 

T±2n*' 

6    "                                            4 

33.       £  ±  2  nr, 

36.       ^_j-2n7r,                        39.    —  ±  2  mr, 

3 

2                                                6 

4  7T 

^±2M,r.                              1T±2»*- 

34.       ^  ±  2  W7r, 

37.       -±2wir,                         42.    x  =  —  ^- 

ANSWERS  25 


43.        30°  =  sin-1  \  =  cos"1 £  \/3  =  tan-1  $  V3  =  cor1  \/3. 
60°  =  sin-1  *  V3  =  cos-1  ^  =  tan"1  V3  =  cof1  ^  >/3. 
90°  =  sin-1 1  =  cos-1 0  =  tan-1  oo  =  cotr1 0. 
45°  =  sin"1 1 V2  =  cos"1 1\/2  =  tan"1 1  =  cotr1 1. 
0°  =  sin'1 0  =  cos-1 1  =  tan'1 0  =  cor1  oo . 
n  180°  =  sin-1 0  =  tan-1 0. 
n  90°  =  cos'1 0  =  cot'1 0. 


ANSWEKS 


Exercise  50 

1. 

A 

— 

36°  12'  14". 

10.     A 

= 

132°  43'. 

16. 

A  = 

74°  22'  18". 

B 

— 

85°  52'  19". 

B 

— 

80°  6'  46". 

B  = 

29°  0'  22". 

c 

= 

84°  20'  30". 

b 

= 

76°  29'  10". 

c  = 

59°  41'  57". 

2. 

a 

— 

26°  19'  48". 

11.      a 

= 

113°  53'  56'. 

17. 

a  = 

69°  6'  12". 

B 

= 

74°  4'  42". 

b 

= 

156°  33'  10". 

b  = 

106°  19'  45". 

b 

= 

51°  47'  41". 

c 

= 

68°  10'  51". 

c  = 

95°  45'  20". 

3. 

a 

= 

44°  36". 

12.     A 

_ 

19°  55'  51". 

18. 

A    

68°  22'  26". 

b 

= 

28°  48'  51". 

A' 

= 

160°  4'  9". 

B  = 

26°  50'  36". 

c 

= 

51°  22'  13". 

a 

= 

13°  44'  24". 

a  — 

35°  17'  40". 

4. 

A 

_ 

76°  20'  45". 

a' 

= 

166°  15'  36". 

a 



71°  34'  7". 

c 

— 

44°  9'  51". 

19. 

B  = 

118°  6'  14". 

b 



46°  47  '50". 

c'  = 

135°  50'  9". 

b  = 

127°  44'  46". 

a  — 

43°  36'  55". 

5. 

A 

— 

51°  2'  30". 

13.     B 

= 

24°  3'  27". 

b 

= 

17°  26'  29". 

b 

_ 

15°  2'  18". 

20. 

A  = 

35°  55'  46". 

c 

— 

63°  27'. 

0  = 

39°  31'  49". 

a  = 

25°  30'  58". 

6. 

B 

a 
b 

= 

70°  5'  13". 
52°  18'  30". 
65°  24'  9". 

B'  = 

C'  = 

155°  56'  33". 
164°  57'  42". 
140°  28'  11". 

c  — 
A'  = 

a'  = 
c'  = 

47°  13'  55". 
144°  4'  14". 
154°  29'  2". 
132°  46'  5". 

7. 

Ji 

= 

65°  9'  27". 

14.     B 

= 

23°  21'  14". 

a 

_ 

118°  6'  23". 

B' 

= 

156°  38'  46". 

21. 

A  = 

7°  54'. 

c 



102°  38'  49". 

b' 

_ 

158°  54'  42". 

b  = 

70°  46'  52". 

8. 

A 



131°  27'  18". 

b 

= 

21°  5'  18". 

c  = 

70°  58'  11". 

B 

_ 

80°  55'  27". 

c  — 

65°  10'  50". 

22. 

B  = 

40°  28'  56". 

c 

= 

98°  6'  42". 

c'  = 

114°  49'  10". 

139°  31'  4". 

9. 

A 

— 

70°  23'  52". 

15.     A  = 

95°  10'  9". 

23. 

c  = 

89°  59'  5". 

a 

— 

54°  40'  14". 

b 

= 

54°  56'  29". 

b 

= 

149°  50'  25". 

c  = 

93°  37'  4". 

24. 

b  = 

8°  40'  30". 

26. 

A 

= 

36.2°.          29. 

A  =  76.35 

o. 

32.     B  = 

65.16°. 

35. 

A  =  132.71 

o 

B 

— 

85.87°. 

a  -71.57°. 

a  = 

118.11°. 

5  =  80.11° 

c 

= 

84.34°. 

b  =  46.8°, 

c  = 

102.65°. 

b  =  76.49° 

. 

27. 

A 

— 

26.33°.         30. 

^1  =  51.03 

o 

33.     A  = 

131.46°. 

36. 

a  =  113.9° 

. 

B 

— 

74.019°. 

b  =  17.44 

o 

B  = 

80.92°. 

b  =  156.53°. 

b 

= 

51.8°. 

c  =  63.44 

o 

c  — 

98.11°. 

c  =  68.18° 

28. 

a 

_ 

44.6°.           31. 

B  =  70.09°. 

34.     A  = 

70.4°. 

37. 

A  =  19.93°. 

b 

= 

28.77°. 

b  =  65.41 

0 

a  = 

54.68°. 

A'=  160.07 

0 

c 

= 

51.37°. 

a  =  52.3°.                    b  = 

149.84°. 

a  =  13.74° 

ANSWERS 


38. 


a' 
c 

=  166.26°. 
=  44.19°. 

b' 
c 

=  158.94°. 
=  65.15°. 

43.     A 
B 

=  68.37°. 
=  26.84°. 

46.      a  =  7.9°. 
6  =  70.78°. 

c' 

=  135.81°. 

c' 

=  114.85°. 

a 

=  35.32°. 

c  =  70.97°. 

B 

=  24.06°. 

40.     A 

=  95.17°. 

44.     B 

=  118.11°. 

47.     B  =  40.49°. 

h 

=  15.04°. 

b 

=  54.94°. 

b 

=  127.75°. 

B'  =  139.51°. 

c, 

=  39.53°. 

c 

=  93.62°. 

c 

=  43.625°. 

=  155.94°. 

41.     A 

=  74.37°. 

45.     A 

=  35.92°. 

48.      c  =  89.985°. 

b' 

=  164.96°, 

B 

=  29°. 

A' 

=  144.08°. 

49.      b  =  8.8°. 

c' 

=  140.47°. 

c 

=  59.7°. 

a 

=  25.61°. 

B 

=  28.35°. 

42.      a 

=  69.1°. 

a' 

=  154.49°. 

B' 

=  156.65°. 

b 

=  106.36°. 

c 

=  47.24°. 

b 

=  21.06°. 

c 

=  95.76°. 

c' 

=  132.76°. 

Exercise  51 


1. 

B 

- 

145°  26'. 

7.  A 

=  38°  4'  46". 

c 

=  103.05°. 

a 

— 

98°  35'  33''. 

c 

=  91°  27  '50". 

A1 

=  38.94°. 

C 

— 

102°  14'  1". 

8.  A 

=  7°  29'  34". 

a' 

=  40.18°. 

2. 

A 

_ 

166°  37'  20". 

A' 

=  172°  30'  26". 

c' 

=  76.95°. 

B 

_ 

139°  10'  16". 

a 

=  80°  50'  30". 

14.  A 

=  37.84°. 

C 

_ 

137°  24'  22". 

a' 

=  99°  9'  30". 

B 

=  37.84°. 

3. 

a 



110°  10'  11". 

10.  B 

=  145.44°. 

C 

=  133°. 

b 

_ 

172°  35'  46". 

a 

=  98.59°. 

15.  C 

=  159.2°. 

C 

= 

106°  25'  11". 

C 

=  102.23°. 

•  c 

=  143.78°. 

4. 

a 

_ 

139°  49'  16". 

11.  A 

=  166.65°. 

16.  A 

=  38.08°. 

A 

_ 

141°  3'  46  '. 

B 

=  139.17°. 

B 

=  38.08°. 

c 

_ 

103°  4'. 

C 

=  137.41°. 

c 

=  91.47°. 

A1 

— 

38°  56'  12". 

A' 

=  13.35°. 

17.  A 

=  7.49°. 

a' 

— 

40°  10'  44". 

B' 

=  40.83°. 

d 

=  80.9°. 

c' 

= 

76°  56'. 

C' 

=  42.59°. 

A' 

=  172.51°. 

5. 

A 

— 

37°  50'  18". 

12.  a 

=  110.17°. 

a' 

=  99.1°. 

C 

= 

133°  3'. 

b 

=  172.55°. 

1} 

= 

37°  50'  18". 

C 

=  106.42°. 

6. 

.C 

= 

159°  12'  12". 

13.  A 

=  141.06°. 

c 

= 

143°  46'  39". 

a 

=  139.82°. 

19.  Tetrahedron:  70° 31' 46" 

=  70.55°. 
Octahedron  :  109°  28'  27" 

=  109.48°. 
Dodecahedron  :   116°  33'  45" 

=  116.6°. 
Icosahedron  :  138°  11'  36" 

=  138.27°. 

22.   Dihedral  angle  between  two  adjacent  faces  =  109°  28'  13"  =  109.47°. 
Dihedral  angle  between  each  face  and  the  base  =  54°  44'  6"  =  54.74°. 


20.  Surface  =  2064.57 

[2064.29]. 
Volume  =  7662.8 
[7668.3J. 

21.  Cot  i  M  -  VCos  m. 


28 


ANSWERS 


23.  Face  angle  at  base  of  frustum  equals  81°  6'  32"  =  81.11°. 
Dihedral  angle  between  two  adjacent  faces  =  91.24°  =  91.4°. 

24.  62°  53'  14"  =  62.89°.        25.    157°  31'  20"  =  157.4°. 


C  =  53°  30'  3". 
a  =  48°  30'  20". 

2.  A=  61°  28' 30". 
C=  42J12'54". 
b  =  94°  41' 17". 

3.  A  =  129°  40'  45". 
5=61°  38'  9". 

c  =  65°  54'. 

4.  B  =  44°  8'  29". 
C  =  133°  51'  51". 
a  =  70°  47°  7". 

5.  B  =  125°  40' 7". 
O  =  34°  9' 47". 
a  =  73°  34'  40". 


Exercise  53 

6.  B  =  162°  38' 21". 
C=  147°  52' 21". 
a  =  77°  8'. 

.7.   A  =  132°  12' 37". 

C  =  64°  49'  57". 
b  =  69°  59'  47". 

8.  B  =  92.04°. 
C  =  53.5°. 
a  =  48.55°. 

9.  A  =  61.47°. 
(7  =  42.21°. 
5  =  94.76°. 

10.   A  =  129.68°. 
B  =  61.64°. 
c  =  65.92°. 


11.  B  =  44.15°. 
C  =  133.86°. 

a  =  70.77°. 

12.  B  =  125.67°. 
0  =  34.17°. 
a  =  73.53°. 

13.  5=162.64°. 

C  =  147.87°. 
a  =  77.13°. 

14.  A  =  132.21°. 
C  =  64.83°. 
B  =  70°. 


Exercise  54 


1. 

A 

=  64°  36'  45". 

6.    .5  =  49°  13'. 

11.    (7 

=  83.76°. 

b 

=  49°  0'  15". 

a  =  67°  49'  54". 

a 

=  123.74°. 

c 

=  36°  41'  37". 

c  =  117°  26'  34". 

b 

=  134.91°. 

2. 

C 

=  157°  52'  54". 

7.    C  =  137°  40'  54". 

12.    A 

=  123.3°. 

a 

=  114°  42'  41". 

a=83°37'28". 

b 

=  70.26°. 

b 

=  39°  3'. 

b  =  45°  12'  20". 

c 

=  145.95°. 

3. 

B 

=  48°  49'  12". 

8.    .4  =  64.6°. 

13.    B 

=  49.25°. 

a 

=  124°  42'  42". 

b  =  49.01°. 

a 

=  67.82°. 

c 

=  103°  23'  42". 

c  =  36.7°. 

c 

=  117.44°. 

4. 

C 

=  83°  39'  16". 

9.    (7=157.88°. 

14.    C 

=  137.68°. 

a 

=  123°  43'  44" 

a  =  114.47°. 

a 

=  83.61°. 

b 

=  134°  55'  16". 

6  =  39.07°. 

b 

=  45.21°. 

5. 

A 

=  123°  18'  15". 

10.    JS  =  48.82°. 

b 

=  70°  15'  24". 

a  =  124.71°. 

c 

=  145°  56'  38". 

c  =  103.4°. 

Exercise  55 

1. 

A 

=  85°  35'  14". 

2.   A  =  143°  3'  48''. 

3.   A-- 

=  34°  15'  4". 

B 

=  49°  35'  34". 

B  =  79°  54'  4". 

B  =  42°  15'  16". 

C 

=  59°  38'  40", 

C  =  55°  3'  4", 

(7  =  121°  36'  20", 

ANSWERS 


29 


4.  4  =  113°  39' 17". 
B  =  123°  40'  19". 
C=  159°  43' 22". 

5.  A  =  110°  51'  20". 
£  =  38°  26 '46". 

C  =  48°  56'  8". 

6.  B  =  151°  44'  47'  . 

7.  C  =128°  53'  9". 


8.  .4  =  85.6°. 
B  =  49.59°. 
(7=59.66°. 

9.  A  =  143.07°. 
B  =  79.91°. 
C  =  55.06°. 

10.    4  =  34.24°. 

B  =  42.25°. 
0=121.61°. 


11.  A  =  113.66°. 
B  =  123.67°. 
O=  159.72°. 

12.  4  =  110.88°. 
B  =  38.43°. 
O  =  48.93°. 

13.  B  =  151.74°. 

14.  O  =  128.88°. 


1.  a  =  147°  23' 29". 
b  =  122°  16'  32". 
c  =  60°  41' 31". 

2.  c  =  126°  58'  19". 

3.  a  =  44°  11'  33". 
6  =  113°  9' 29". 
c  =  113°  9'  29". 

4.  a  =  73°  57'  28". 

5.  a  =  71°  8' 55". 
b  =  75°  12'. 

c  =  59°  58'  4". 


Exercise  56 

6.  b  =  102°  46'  10". 

7.  a  =27°  31' 12". 
5  =  86°  14'  34". 
c  =  83°  31'  42". 

8.  c  =  146°  37' 16". 

10.  a  =  147.39°. 
b  =  122.29°. 
c  =  60.69°. 

11.  c  =  126.97°. 

12.  a  =  44.2°. 

b  =  113.16°. 
c=  113.16°. 


13.  a  =  73.96°. 

14.  a  =  71.14°. 

b  =  75.2°. 
c  =  59.98°. 

15.  b  =  102.79°. 

16.  a  =  27.52°. 
b  =  86.24°. 
c  =  83.53°. 

17.  c  =  146.61°. 


1.  B  =  39°  48'  30". 
0  =  50°  30'  44". 
c  =  40°  30'  3". 

2.  .B  =  55°  52' 40". 

O  =  20°  9' 48". 
c  =  20°  16'  30". 

3.  4  =  115°  57' 58". 
B  =  57°  34'  53". 

a  =  95°  18'  14". 
A'  =  25°  44'  34". 
B'  =  122°  25'  8". 

a'  =  28°  45'  6". 

4.  5  =  56°  29'  13". 
O=  136°  31' 8". 
c  =  126°  1' 22". 

S'  =  123°  30' 47". 
O'  =  14°  34'. 
c'  =  17°  11' 42", 


Exercise  57 

5.  a  =  67°  3'  48". 

6.  O  =  40°  24' 30". 
0'  =  139°  35'  30". 

7.  B  =  74°  7'. 

4  =  80°  24'  46". 
a  =  117°  37 '23". 
B'  =  105°  53'. 
4' =  37°  31' 29". 
«'  =  146°  48' 37". 

8.  £  =  39.8°. 
0  =  50.5°. 
c  =  40.5°. 

9.  5  =  55.88°. 
O  =  20.17°. 
c  =  20.28°. 

10.    4=115.98°. 
B  =  57.56°, 


a  =  95.33°. 
A'  =  25.69°. 
B'  =  122.44°. 

a'  =  28.7°. 

11.  O=  83.76°. 
a  =  113.74°. 
b  =  134.91°. 

12.  4  =  67.1°. 

13.  O  =  40.39°. 
C'  =  139.61°. 

14.  4  =  80.37°. 
B  =  74.15°. 

a  =  117.66°. 
4'  =  37.59°. 
.B'  =  105.85°, 

a'  =  146.8°. 


30 


1. 


2. 


B  -  57°  37'  36'  . 
b  =  34°  34'  56". 
c  =  35°  35'  56". 

C  =  73°  24'  50". 
b  =  33°  27'  6". 
c  =  42°  14'  44". 


3.  4  =  136°  51'  12". 

a  =  128°  19'  56". 
b  =  143°  32'  40". 

4.  B  =  143°  40'  5". 

6  -  157°  33'  6". 

c  =  39°  24'. 
B'  =  59°  12'  46". 
&'  =  33°37'14". 
c'  =  140°  36'. 

5.  B  =  115°  58'  30". 

a  =  54°  21'  7". 


6. 


8. 


10. 


11. 


ANSWERS 
Exercise  58 


^'  =  162°  48' 34". 
a'  =  125°  38'  53". 
b'  =161°  16' 50''. 

C  =  33°  36'  24". 

b  =  21°  13'  16". 

c  =  13°  54'  18". 

a  =  66°  59' 50''. 
a' =  113°  0' 10". 
5  =  57.59°. 

6  =  34.57°. 

c  =  35.6°. 

C  =  73.42°. 

b  =  33.45°. 

c  =  42.25°. 
A  =  136.85°. 

a  =  128.34°. 

6  =  143.54°. 


12. 


13. 


14. 


15. 


B  =  143.63°. 

b  =  157.53°. 

c  =  39.41°. 
B1  =  59.21°. 
b'  =  33.63°. 
c'  =  140.59°. 

B  =  115.94°. 

a  =  54.34°. 

b  =  77.68°. 
B'  =  162.82°. 
a'  =  125.66°. 
b'  =  161.27°. 

C  =  33.62°. 
b  =  21.22°. 
c  =  13.91°. 

a  =  67°. 
a'  =  113°. 


7.    19.505  sq.  in. 
19.503  sq.  in. 


Exercise  59 

In  this  exercise,  where  two  answers  are  given  to  an  example,  the  first  answer  is 
obtained  by  use  of  five-place  tables  and  the  second  answer  by  use  of  four-place 
tables. 

1.  2.513  sq.  ft.  4.    254.82  sq.  in. 

2.  13.548  ft.  5.    43,793  sq.  mi. 
13.547  ft.  43,780  sq.  mi. 

3.  17.279  sq.  mi.  6.   4,379,300  sq.  mi. 
17.265  sq.  mi.  4,379,000  sq.  mi. 

8.   Each  angle  =  62°  23'  22" 

=  62.39°. 

Perimeter    =  6273.42  statute  miles 
=  6277.14  statute  miles. 

9.    137.439  sq.  ft.  11.    827.96  sq.  m.  13.    8008  sq.  m. 

10.    195.36  sq.  in.  12.    18.767  sq.  in.  14.    2547.53  sq.  ft. 

15.   4867.33  sq.  in.  =  area  of  1st  triangle. 
1135  sq.  in.        =  area  of  2d  triangle. 

16.  137.4  sq.  ft.  18.    827.8  sq.  m.  20.    8006  sq.  m. 

17,  195.3  sq.  in.  19.    18.76  sq.  in.  21.    2547.65  sq.  ft. 

22.   4858.9  sq.  in.  =  area  1st  triangle. 
1134.2  sq.  in.  =  area  2d  triangle. 


ANSWERS  31 


Exercise  60 

In  this  exercise,  where  two  answers  are  given  to  an  example,  the  first  answer  is 
obtained  by  use  of  five-place  tables  and  the  second  answer  by  use  of  four-place 
tables. 

1.  Distance  =  6485.5  nautical  miles. 

6484.8  nautical  miles. 

N.  47°  48'  37"  W.  =  bearing  of  Halifax  from  Cape  Town. 
S.  59°  48'  34"  E.  =  bearing  of  Cape  Town  from  Halifax. 

2.  Lat.  42°  9'  36"  N.  =  42. 1°  N.  4.    Lat.  =  53°  25'  47"  N. 
Lon.  70°  8'  38"  W.  =  70. 14°  W.  53.43°  N. 

3.  Lon.  =  13°  11'  37"  W.  Distance  =  3f 2'7  statute  mi^ 

13  2°  W  3301.5  statute  miles. 

Distance  =  3113.64  statute  miles. 
3112.86  statute  miles. 

5.  Distance  =  2080.5  nautical  miles. 

2082  nautical  miles. 

N.  53°  38'  44"  E.  =  bearing  of  San  Francisco  from  Honolulu. 
N.  53.65°  E.  =  bearing  of  San  Francisco  from  Honolulu. 
S.  71° 44'  14"  W.—  bearing  of  Honolulu  from  San  Francisco. 
S.  71.75°  W.  =  bearing  of  Honolulu  from  San  Francisco. 

6.  Lat.  =  33°  40' 21  "N. 

=  33.67°  N. 

7.  Length  of  arc*  of  great  circle  =  1688.1  nautical  miles 

=  1888.4  nautical  miles. 

Length  of  parallel  of  latitude  —  1700.6  nautical  miles 
=  1701  nautical  miles. 

8.  Distance  =  5769.43  nautical  miles 

=  5769  nautical  miles. 

N.  60°  24'  50"  W.  =  bearing  of  Manila  from  Seattle. 
N.  60.42°  W.  =  bearing  of  Manila  from  Seattle. 

9.  Lat.  =  53°  52'  47"  N.  15.    (1)  2641.2  statute  miles.  • 

=  53.88°  N.  2641.25  statute  miles. 

Long.  =  152°  38'  W.  (2)  2950.6  statute  miles. 

=  152.63°  W.  2950.67  statute  miles. 

10.  7  hours  41  minutes  43  seconds  A.M.  (3)  2948  statute  miles. 

7  hours  41  minutes  24  seconds  A.M.  (4)  18539.17  statute  miles. 

18535  statute  miles. 

11.  4  hours  41  minutes  A.M. 

16.  2  hours  7  minutes  P.M. 

12.  5  hours  27  minutes  P.M. 

17.  14  hours  51  minutes  10  seconds. 

14  hours  51  minutes  10  seconds. 
121.87°. 

14.  6  hours  14  minutes  A.M. 


32  ANSWERS 

18.  Angle  TOP  =  37°  17'  14"  * 

=  37.29°. 

OP  makes  with  the  plane  XO  T  an  angle  =  30°. 
OP  makes  with  the  plane  XOZ  an  angle  =  52°  42'  46". 
OP  makes  with  the  plane  ZO  Y  an  angle  =  20°. 

19.  Length  of  perpendicular  from  Pto  OX  =  17.173 

=  17.175. 

Length  of  perpendicular  from  Pto  OT=  11.072. 
Length  of  perpendiculer  from  Pto  OZ  =  15.827. 
Length  of  projection  of  OP  on  OX—  6.254 

=  6.251. 

Length  of  projection  of  OP  on  OY=  14.54. 
Length  of  projection  of  OP  on  OZ  =  9.138. 

20.  Length  of  perpendicular  from  P  to  plane  XOY  =  9.138. 
Length  of  perpendicular  from  Pto  plane  XOZ  =  14.54. 
Length  of  perpendicular  from  P  to  plane  YOZ  =  6.25. 
Length  of  projection  of  OP  on  plane  XOY  =  15.827. 
Length  of  projection  of  OP  on  plane  XOZ  =  11.072. 
Length  of.  projection  of  OP  on  plane  YOZ-—  17.175. 

21.  OP  makes  with  the  plane  XO  Y  an  angle  =  22°  52'  42" 

=  22.88°. 

Cutting  plane  makes  with  OX  an  angle  =  33°. 
Cutting  plane  makes  with  O  Y  an  angle  =  48°. 
Cutting  plane  makes  with  OZ  an  angle  =  22°  52'  42" 

=  22.88°. 
Cutting  plane  makes  with  plane  XOY&n  angle  =  67°  7'  *18" 

=  67.12° 

Cutting  plane  makes  with  plane  XOZ  an  angle  =  42°. 
Cutting  plane  makes  with  plane  YOZ  an  angle  =  57°. 

25.   If  D  represents  the  diagonal  of  the  parallelepiped,  then 

D  =  a?  +  62  +  c2  +  2  ab  cos  7  +  2  be  cos  a  +  2  ac  cos  0. 


w  A/ 

tyL    ( 

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